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Magnetic sounding using data collected from the Juno mission can be used to provide constraints on Jupiter's interior. However, inwards continuation of reconstructions assuming zero electrical conductivity and a representation in spherical harmonics are limited by the enhancement of noise at small scales. Here we describe new reconstructions of Jupiter's internal magnetic field based on physics-informed neural networks and either the first 33 (PINN33) or the first 50 (PINN50) of Juno's orbits. The method can resolve local structures, and allows for weak ambient electrical currents. Our models are not hampered by noise amplification at depth, and offer a much clearer picture of the interior structure. We estimate that the dynamo boundary is at a fractional radius of 0.8. At this depth, the magnetic field is arranged into longitudinal bands, and strong local features such as the great blue spot appear to be rooted in neighbouring structures of oppositely signed flux.

Marchenko focusing functions are in their essence wavefields that satisfy the wave equation subject to a set of boundary, initial, and focusing conditions. Here, we show how Marchenko focusing functions can be modeled by finding the solution to a wavefield reconstruction inversion problem. Our solution yields all elements of the focusing function including evanescent, refracted, and trapped waves (tunneling). Our examples indicate that focusing function solutions in higher dimensions are however challenging to compute by numerical means due to the appearance of strong evanescent waves., Comment: This document constitutes a summary of results in the PhD Transfer Report of R.F. Hajjaj, successfully defended on October 19th, 2022

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 to its present speed of 50-60 km/yr. 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. Localised modelling shows that elongation of the Canadian lobe, likely caused by an alteration in the pattern of core-flow between 1970 and 1999, significantly 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.

Mounting evidence from both seismology and experiments on core composition suggests the existence of a layer of stably stratified fluid at the top of Earth's outer core. In this work we examine the structure of the geomagnetic field within such a layer, building on the important but little known work of Malkus (1979). We assume (i) an idealised magnetostrophic spherical model of the geodynamo neglecting inertia, viscosity and the solid inner core, and (ii) a strongly stratified layer of constant depth immediately below the outer boundary within which there is no spherically radial flow. Due to the restricted dynamics, Malkus showed that the geomagnetic field must obey a certain condition which is a more restrictive version of the condition of Taylor (1963). The nonlinear nature of these constraints makes finding a magnetic field that obeys them, here termed a Malkus state, a challenging task. Nevertheless, such Malkus states when constrained further by geomagnetic observations have the potential to probe the interior of the core. By focusing on a particular class of magnetic fields for which the Malkus constraints are linear, we describe a constructive method that turns any purely-poloidal field into an exact Malkus state by adding a suitable toroidal field. We consider poloidal fields following a prescribed smooth profile within the core that match observation-derived models of the magnetic field in either epoch 2015 or the 10000-yr time averaged field. Multiple possible solutions for the toroidal field exist, hence we determine the Malkus state of miumum toroidal energy and we find that it has a strong azimuthal toroidal field, larger than the observed poloidal component at the core-mantle boundary. For the 2015 field for a layer of depth 300 km, we estimate a root mean squared azimuthal toroidal field of 3 mT with a pointwise maximum of 8 mT occurring at a depth of about 70 km., Comment: 27 pages, 10 figures

Motivated by the dynamics within terrestrial bodies, we consider a rotating, strongly thermally stratified fluid within a spherical shell subject to a prescribed laterally inhomogeneous heat-flux condition at the outer boundary. Using a numerical model, we explore a broad range of three key dimensionless numbers: a thermal stratification parameter (the relative size of boundary temperature gradients to imposed vertical temperature gradients), $10^{-3} \le S \le 10^{4}$, a buoyancy parameter (the strength of applied boundary heat flux anomalies), $ 10^{-3} \le B \le 10^{6}$, and the Ekman number (ratio of viscous to Coriolis forces), $10^{-6} \le E \le 10^{-4}$. We find both steady and time-dependent solutions and delineate the temporal regime boundaries. We focus on steady-state solutions, for which a clear transition is found between a low $S$ regime, in which buoyancy dominates dynamics, and a high $S$ regime, in which stratification dominates. For the latter case, the radial and horizontal velocities scale respectively as $u_r \sim S^{-1}$, $u_h \sim S^{-\frac{3}{4}}\ B^{\frac{1}{4}}$ and are confined to boundary-induced flow within a thin layer of depth $(S\ B)^{-\frac{1}{4}}$ at the outer edge of the domain. For the Earth, if lower-mantle heterogeneous structure is due principally to chemical anomalies, we estimate that the core is in the high-$S$ regime and steady flows arising from strong outer-boundary thermal anomalies cannot penetrate the stable layer. However, if the mantle heterogeneities are due to thermal anomalies and the heat-flux variation is large, the core will be in a low-$S$ regime in which the stable layer is likely penetrated by boundary-driven flows.

In his seminal work, Taylor (1963) argued that the geophysically relevant limit for dynamo action within the outer core is one of negligibly small inertia and viscosity in the magnetohydrodynamic equations. Within this limit, he showed the existence of a necessary condition, now well known as Taylor's constraint, which requires that the cylindrically-averaged Lorentz torque must everywhere vanish; magnetic fields that satisfy this condition are termed Taylor states. Taylor further showed that the requirement of this constraint being continuously satisfied through time prescribes the evolution of the geostrophic flow, the cylindrically-averaged azimuthal flow. We show that Taylor's original prescription for the geostrophic flow, as satisfying a given second order ordinary differential equation, is only valid for a small subset of Taylor states. An incomplete treatment of the boundary conditions renders his equation generally incorrect. Here, by taking proper account of the boundaries, we describe a generalisation of Taylor's method that enables correct evaluation of the instantaneous geostrophic flow for any 3D Taylor state. We present the first full-sphere examples of geostrophic flows driven by non-axisymmetric Taylor states. Although in axisymmetry the geostrophic flow admits a mild logarithmic singularity on the rotation axis, in the fully 3D case we show that this is absent and indeed the geostrophic flow appears to be everywhere regular., Comment: 29 Pages, 8 figures

In investigating the properties of a certain class of homogeneous polynomials, we discovered an identity satisfied by their coefficients which involves simple 2F1 Gauss hypergeometric functions. This result appears to be new and we supply a direct proof. The simplicity of the identity is suggestive of a deeper result.