3 results
Subglacial sediment distribution from constrained seismic inversion, using MuLTI software: examples from Midtdalsbreen, Norway
- Siobhan F. Killingbeck, Adam D. Booth, Philip W. Livermore, Landis J. West, Benedict T. I. Reinardy, Atle Nesje
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- Journal:
- Annals of Glaciology / Volume 60 / Issue 79 / September 2019
- Published online by Cambridge University Press:
- 07 May 2019, pp. 206-219
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Fast ice flow is associated with the deformation of subglacial sediment. Seismic shear velocities, Vs, increase with the rigidity of material and hence can be used to distinguish soft sediment from hard bedrock substrates. Depth profiles of Vs can be obtained from inversions of Rayleigh wave dispersion curves, from passive or active-sources, but these can be highly ambiguous and lack depth sensitivity. Our novel Bayesian transdimensional algorithm, MuLTI, circumvents these issues by adding independent depth constraints to the inversion, also allowing comprehensive uncertainty analysis. We apply MuLTI to the inversion of a Rayleigh wave dataset, acquired using active-source (Multichannel Analysis of Surface Waves) techniques, to characterise sediment distribution beneath the frontal margin of Midtdalsbreen, an outlet of Norway's Hardangerjøkulen ice cap. Ice thickness (0–20 m) is constrained using co-located GPR data. Outputs from MuLTI suggest that partly-frozen sediment (Vs 500–1000 m s−1), overlying bedrock (Vs 2000–2500 m s−1), is present in patches with a thickness of ~4 m, although this approaches the resolvable limit of our Rayleigh wave frequencies (14–100 Hz). Uncertainties immediately beneath the glacier bed are <280 m s−1, implying that MuLTI cannot only distinguish bedrock and sediment substrates but does so with an accuracy sufficient for resolving variations in sediment properties.
Penetration of boundary-driven flows into a rotating spherical thermally stratified fluid
- Grace A. Cox, Christopher J. Davies, Philip W. Livermore, James Singleton
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- Journal:
- Journal of Fluid Mechanics / Volume 864 / 10 April 2019
- Published online by Cambridge University Press:
- 11 February 2019, pp. 519-553
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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), $10^{-3}\leqslant S\leqslant 10^{4}$, a buoyancy parameter (the strength of applied boundary heat-flux anomalies), $10^{-2}\leqslant B\leqslant 10^{6}$, and the Ekman number (ratio of viscous to Coriolis forces), $10^{-6}\leqslant E\leqslant 10^{-4}$. We find both steady and time-dependent solutions and delineate the regime boundaries. We focus on steady-state solutions, for which a clear transition is found between a low $S$ regime, in which buoyancy dominates the dynamics, and a high $S$ regime, in which stratification dominates. For the low-$S$ regime, we find that the characteristic flow speed scales as $B^{2/3}$, whereas for high-$S$, the radial and horizontal velocities scale respectively as $u_{r}\sim S^{-1}$, $u_{h}\sim S^{-3/4}B^{1/4}$ and are confined within a thin layer of depth $(SB)^{-1/4}$ 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.
Taylor state dynamos found by optimal control: axisymmetric examples
- Kuan Li, Andrew Jackson, Philip W. Livermore
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- Journal:
- Journal of Fluid Mechanics / Volume 853 / 25 October 2018
- Published online by Cambridge University Press:
- 29 August 2018, pp. 647-697
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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 $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-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 $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects are invoked, as well as new distributions. One such new $\unicode[STIX]{x1D6FC}$-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.