Convection Driven Dynamos in Rotating Spheres

Of the objects in the solar system the Earth, Mercury, Jupiter, Saturn, Uranus, Neptune, Ganymede, and the Sun exhibit a magnetic field. These magnetic fields are believed to be generated by the magnetohydrodynamic dynamo process, in which current, generated as electrically conducting fluid crosses magnetic field lines, regenerates the magnetic field. Although most of the bodies listed above are believed to consist of a fluid outer core with a solid inner core, i.e. a spherical shell geometry, the full sphere dynamo problem is of physical interest as the dynamo of the early Earth, the ancient dynamo of Mars, and possibly Venus, the Moon and (currently) Mercury, are believed to have had no solid inner core. In this thesis we consider numerically the problem of magnetic field generation in a full sphere of rotating uniformly conducting fluid driven by a volumetric heat source. In order to numerically integrate the governing system of equations we combine the poloidal-toroidal field representation of Elsasser (1946) and Bullard&Gellman (1954) with an implicit/explicit multi-step Gear timestepping method and finite differences in radius. For the implicit radial differencing we develop a generalised compact finite-difference method which results in high order/low bandwidth timestepping systems, and we demonstrate that this method is competitive with other finite-difference methods: standard finite differences, Padé finite-differences, and the combined compact finitedifference schemes of Chu&Fan (1998). The numerical integrator is applied to three physical problems of interest. The first is kinematic dynamo action in a sphere. We investigate the possibility of dynamo action for flows with a missing component in spherical polar coordinates and find the growth rates are highly sensitive to changes in the truncation level. Nevertheless, we do find a working kinematic dynamo with axisymmetric velocity with no azimuthal component which demonstrates convincing convergence. The second problem we consider is that of thermal convection in the absence of a magnetic field in a rotating sphere. We fix the Ekman and Prandtl number (E; Pr) = (5 10¿4; 0:7) and obtain an estimate of the critical Rayleigh number Rac for the onset of convection, and describe the main characteristic of the flow for the convection solutions for Ra 1:4 Rac and Ra 5 Rac. These solutions are primarily for comparison for solutions computed in the third problem: dynamical dynamo action in a rotating sphere. The primary aim is to survey dynamo solutions for the fixed Ekman and Prandtl numbers (E; Pr) = (5 10¿4; 0:7), for magnetic Prandtl number varied from 1 to 40 and the modified Rayleigh number varied up to a few times the critical value for the onset of convection. We consider the solutions through the lens of dynamo scaling laws, but find no universally satisfactory theoretical or numerical scaling law. We also consider a weak/strong field classification of the solutions, finding highly localised force balances. We finish by considering three solutions in detail which represent three distinct classes of dynamo solution: an oscillating dipolar solution, an oscillating quadrupolar solution and a chaotic solution which oscillates between two different hemispherical states. Finally, we develop a first approach to the problem of dynamo action in a fluid sphere as it cools (with no internal heat source), and we present some first convective solutions which function exactly as we expect: the convection dieing down as the fluid cools

pyC2^2Ray: A flexible and GPU-accelerated Radiative Transfer Framework for Simulating the Cosmic Epoch of Reionization

Detailed modelling of the evolution of neutral hydrogen in the intergalactic medium during the Epoch of Reionization, $5 \leq z \leq 20$, is critical in interpreting the cosmological signals from current and upcoming 21-cm experiments such as Low-Frequency Array (LOFAR) and the Square Kilometre Array (SKA). Numerical radiative transfer codes offer the most physically motivated approach for simulating the reionization process. However, they are computationally expensive as they must encompass enormous cosmological volumes while accurately capturing astrophysical processes occurring at small scales ($\lesssim\rm Mpc$). Here, we present pyC$^2$Ray, an updated version of the massively parallel ray-tracing and chemistry code, C$^2$Ray, which has been extensively employed in reionization simulations. The most time-consuming part of the code is calculating the hydrogen column density along the path of the ionizing photons. Here, we present the Accelerated Short-characteristics Octhaedral RAytracing (ASORA) method, a ray-tracing algorithm specifically designed to run on graphical processing units (GPUs). We include a modern Python interface, allowing easy and customized use of the code without compromising computational efficiency. We test pyC$^2$Ray on a series of standard ray-tracing tests and a complete cosmological simulation with volume size $(349\,\rm Mpc)^3$, mesh size of $250^3$ and approximately $10^6$ sources. Compared to the original code, pyC$^2$Ray achieves the same results with negligible fractional differences, $\sim 10^{-5}$, and a speedup factor of two orders of magnitude. Benchmark analysis shows that ASORA takes a few nanoseconds per source per voxel and scales linearly for an increasing number of sources and voxels within the ray-tracing radii.Comment: 16 pages, 13 figure

Magnetic effects on the low-T/|W| instability in differentially rotating neutron stars

Dynamical instabilities in protoneutron stars may produce gravitational waves whose observation could shed light on the physics of core-collapse supernovae. When born with sufficient differential rotation, these stars are susceptible to a shear instability (the "low-T/|W| instability"), but such rotation can also amplify magnetic fields to strengths where they have a considerable impact on the dynamics of the stellar matter. Using a new magnetohydrodynamics module for the Spectral Einstein Code, we have simulated a differentially-rotating neutron star in full 3D to study the effects of magnetic fields on this instability. Though strong toroidal fields were predicted to suppress the low-T/|W| instability, we find that they do so only in a small range of field strengths. Below 4e13 G, poloidal seed fields do not wind up fast enough to have an effect before the instability saturates, while above 5e14 G, magnetic instabilities can actually amplify a global quadrupole mode (this threshold may be even lower in reality, as small-scale magnetic instabilities remain difficult to resolve numerically). Thus, the prospects for observing gravitational waves from such systems are not in fact diminished over most of the magnetic parameter space. Additionally, we report that the detailed development of the low-T/|W| instability, including its growth rate, depends strongly on the particular numerical methods used. The high-order methods we employ suggest that growth might be considerably slower than found in some previous simulations.Comment: REVTeX 4.1, 21 pages, 18 figures, submitting to Physical Review

Strong shock wave generation by fast electron energy deposition in shock ignition relevant plasmas

The potential role of fast electrons is one of the major unknowns in shock ignition inertial con- finement fusion. Of particular interest is the possibility that they may play a beneficial role in the generation of the ignitor shock by contributing to the ablation pressure. Here, some of the fundamental relations governing fast electron driven shock wave generation in dense plasmas are determined. To that end, a 1D planar hybrid model of fast electron transport through dense plasmas is presented. It is found that, using quasi-realistic electron populations, it is possible to generate shock waves with peak pressures that agree with a simple scaling law and have sustained shock pressures of several hundred Mbars. However, the spatial and temporal scales required for shock waves to fully develop increase with fast electron temperature and can become significant. Careful consideration of this effect is needed when assessing their usefulness as shock wave drivers. A characteristic time of shock wave formation is reinterpreted as the definitive time taken for a localised source of internal energy in an otherwise uniform fluid to drive a blast wave containing its maximum kinetic energy. This relation is of utility in inertial confinement fusion where ignition relies on the conversion of kinetic energy to internal energy at implosion stagnation. However, it is not straightforwardly reproducible by fast electron heating, which highlights the difficulties that may be encountered if fine control over shock wave formation is required. The shape of the density profile seems to be of secondary importance when compared with the consequences of heating using hotter electron populations. When heating times are on the time scale of the ignitor pulse, the density profile affects the efficiency of shock wave formation by determining the transition from an explosive regime to a driven regime of shock wave forma- tion. However, the time taken for the shock wave to contain its maximum kinetic energy is not significantly affected. It is shown that an externally applied magnetic field can constrain the range of fast electrons in solid density planar plastic targets, and enhance localised energy deposition. This mitigates the need for significant spatial and temporal scales when using fast electron populations with extended energy distributions to drive shock waves. However, this comes at the expense of the strength of the shock wave

Advanced numerical approaches in the dynamics of relativistic flows

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