Three-dimensional solutions for the geostrophic flow in the Earth's core

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 figure

Magnetostrophic analysis of Earth's internal magnetic field

Throughout this thesis we build on the central tenet of the seminal work by Taylor (1963), which argued that the geophysically relevant limit for dynamo action within the Earth's outer core is one of negligibly small inertia and viscosity in the magnetohydrodynamic equations. Within this 'magnetostrophic' 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'. We extend the use of this condition, to analyse the geomagnetic field and investigate the underlying geodynamo process within Earth's core, through several key strands of work. Firstly, we detail a general method that is the first to enable correct evaluation of the instantaneous geostrophic flow for any 3D Taylor state, fully incorporating all necessary boundary conditions. Secondly, we explore the subsequent dynamics of Taylor state magnetic fields, calculating the field induced by these flows and hence the rate of change of magnetic field. Importantly, we note the similarities and differences that arise between these magnetostrophic dynamo models and observationally derived geomagnetic field models. We show that Taylor state magnetic fields that remain stable over geophysical time scales are very rare. Thirdly, we consider the prospect of the fluid in the outermost part of Earth's core being stratified. This leads to a necessary adaptation to the Taylor constraint, resulting in the analogous condition within a stratified fluid, termed the 'Malkus constraint'. Implementing this additional constraint allows us to construct a model for the entirety of Earth's outer core, matching observational geomagnetic field models at the core surface, obeying the Malkus constraint in the stratified layer and satisfying the Taylor constraint in the bulk of the core. The results from this model suggest that the dynamics within the stratified layer may be distinct from the inner convective part of the core, characterised not only by suppressed radial flow but by a strong magnetic field. The present-day toroidal field strength immediately beneath the CMB is estimated to be significantly stronger than that within the convective region of the outer core

The use of history to identify anterior cruciate ligament injuries in the acute trauma setting: the 'LIMP index'

YesObjective To identify the injury history features reported by patients with ACL injuries and determine whether history may be used to identify patients requiring follow-up appointments from acute trauma services. Multi-site cross-sectional service evaluation using a survey questionnaire design conducted in the UK. The four injury history features investigated (LIMP) were ‘Leg giving way at the time of injury’, ‘Inability to continue activity immediately following injury’, ‘Marked effusion’ and ‘Pop (heard or felt) at the time of injury’. 194 patients with ACL injury were identified of which 165 (85.5%) attended an acute trauma service. Data on delay was available for 163 (98.8%) of these patients of which 120 (73.6%) had a follow-up appointment arranged. Patients who had a follow-up appointment arranged waited significantly less time for a correct diagnosis (geometric mean 29 vs 198 days; p<0.001) and to see a specialist consultant (geometric mean 61 vs 328 days; p<0.001). Using a referral threshold of any 2 of the 4 LIMP injury history features investigated, 95.8% of patients would have had a follow-up appointment arranged. Findings support the value of questioning patients on specific injury history features in identifying patients who may have suffered ACL injury. Using a threshold of 2 or more of the 4 LIMP history features investigated would have reduced the percentage of patients inappropriately discharged by 22.2%. Evidence presented suggests that this would significantly reduce the time to diagnosis and specialist consultation minimising the chance of secondary complications

Quality control parameters on a large dataset of regionally dissected human control brains for whole genome expression studies

We are building an open-access database of regional human brain expression designed to allow the genome-wide assessment of genetic variability on expression. Array and RNA sequencing technologies make assessment of genome-wide expression possible. Human brain tissue is a challenging source for this work because it can only be obtained several and variable hours post-mortem and after varying agonal states. These variables alter RNA integrity in a complex manner. In this report, we assess the effect of post-mortem delay, agonal state and age on gene expression, and the utility of pH and RNA integrity number as predictors of gene expression as measured on 1266 Affymetrix Exon Arrays. We assessed the accuracy of the array data using QuantiGene, as an independent non-PCR-based method. These quality control parameters will allow database users to assess data accuracy. We report that within the parameters of this study post-mortem delay, agonal state and age have little impact on array quality, array data are robust to variable RNA integrity, and brain pH has only a small effect on array performance. QuantiGene gave very similar expression profiles as array data. This study is the first step in our initiative to make human, regional brain expression freely available

High-Spectral Resolution Dark Holes: Concept, Results, and Promise

Next generation high contrast imaging instruments face a challenging trade off: they will be required to deliver data with high spectral resolution at a relatively fast cadence (minutes) and across a wide field of view (arcseconds). For instruments that employ focal plane wavefront sensing and therefore require super-Nyquist sampling, these requirements cannot simultaneously be met with a traditional lenslet integral field spectrograph (IFU). For the SPIDERS pathfinder instrument, we are demonstrating an imaging Fourier transform spectrograph (IFTS) that offers a different set of tradeoffs than a lenslet IFU, delivering up to R20,000 spectral resolution across a dark hole. We present preliminary results from the SPIDERS IFTS including a chromaticity analysis of its dark hole and demonstrate a spectral differential imaging (SDI) improvement of up to 40 $\times$, and a first application of spectro-coherent differential imaging, combining both coherent differential imaging (CDI) and SDI.Comment: Submitted to the proceedings of adaptive optics for extremely large telescopes

Quantum Sign Permutation Polytopes

Convex polytopes are convex hulls of point sets in the $n$-dimensional space \E^n that generalize 2-dimensional convex polygons and 3-dimensional convex polyhedra. We concentrate on the class of $n$-dimensional polytopes in \E^n called sign permutation polytopes. We characterize sign permutation polytopes before relating their construction to constructions over the space of quantum density matrices. Finally, we consider the problem of state identification and show how sign permutation polytopes may be useful in addressing issues of robustness

On SIC-POVMs in Prime Dimensions

The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, almost every known SIC-POVM fiducial vector is an eigenstate of a "canonical" unitary in the Clifford group. I show that every canonical unitary in prime dimensions p > 3 lies in the same conjugacy class of the Clifford group and give a class representative for all such dimensions. It follows that if even one such SIC-POVM fiducial vector is an eigenvector of such a unitary, then all of them are (for a given such dimension). I also conjecture that in all dimensions d, the number of conjugacy classes is bounded above by 3 and depends only on d mod 9, and I support this claim with computer computations in all dimensions < 48.Comment: 6 pages, no figures. v3 Refs added, improved discussion of previous work. Ref to a proof of the main conjecture also adde

Anti-Adherent Biomaterials for Prevention of Catheter Biofouling

Medical device-associated infections present a leading global healthcare challenge, and effective strategies to prevent infections are urgently required. Herein, we present an innovative anti-adherent hydrogel copolymer as a candidate catheter coating with complementary hydrophobic drug-carrying and eluting capacities. The amphiphilic block copolymer, Poloxamer 188, was chemically-derivatized with methacryloyl moieties and copolymerized with the hydrogel monomer, 2-hydroxyethyl methacrylate. Performance of the synthesized copolymers was evaluated in terms of equilibrium swelling, surface water wettability, mechanical integrity, resistance to encrustation and bacterial adherence, and ability to control release of the loaded fluoroquinolone antibiotic, ofloxacin. The developed matrices were able to provide significant protection from fouling, with observed reductions of over 90% in both adherence of the common urinary pathogen Escherichia coli and encrusting crystalline deposits of calcium and magnesium salts relative to the commonly employed hydrogel, poly (hydroxyethyl methacrylate). Additionally, the release kinetics of a loaded hydrophobic drug could be readily tuned through facile manipulation of polymer composition. This combinatorial approach shows significant promise in the development of suitable systems for prevention of catheter-associated infections