arXiv subjects

Calum S. Skene

Publications and source records attributed to Calum S. Skene.

5 recordsLinked to original sources

arXiv2025

Three-dimensional numerical simulations of neutron star cores in the two-fluid MHD approximation: simple configurations

Magnetic field evolution in neutron star cores is not fully understood. We describe the field evolution both for one barotropic fluid as well as two collisionally coupled barotropic fluids with different density profiles using the anelastic approximation and the Navier-Stokes equations to simulate the evolution in three dimensions. In the one-fluid case, a single fluid describes the motion of the charged particles. In the two-fluid model, the neutral fluid is coupled to the electrically conductive fluid by collisions, the latter being dragged by the magnetic field. In this model, both fluids have distinct density profiles. This forces them to move at slightly different velocities, resulting in a relative motion between the two barotropic fluids -- ambipolar diffusion. We develop a code based on Dedalus and study the evolution of simple poloidal dipolar and toroidal magnetic fields. Previous 2D studies found that poloidal magnetic fields evolve towards a stable Grad-Shafranov equilibrium. In our 3D simulations we find an instability of the two-fluid system similar to the one in the barotropic fluid system. After the instability saturates, a highly non-linear Lorentz force introduces small-scale fluid motion that leads to turbulence, development of a cascade and significant, non-axially symmetric changes in the magnetic field configuration. Fluid viscosity plays an essential role in regularizing the small-scale fluid motion, providing an energy drain.

astro-ph.HE↗

arXiv2025

On nonlinear transitions, minimal seeds and exact solutions for the geodynamo

Nearly fifty years ago, Roberts (1978) postulated that Earth's magnetic field, which is generated by turbulent motions of liquid metal in its outer core, likely results from a subcritical (finite-amplitude) dynamo instability characterised by a dominant balance between Coriolis, pressure and Lorentz forces. Here we numerically explore subcritical convective dynamo action in a spherical shell, using techniques from optimal control and dynamical systems theory to uncover the nonlinear dynamics of magnetic field generation. Through nonlinear optimisation, via direct-adjoint looping, we identify the minimal seed - the smallest magnetic field that attracts to a nonlinear dynamo solution. Additionally, using the Newton-hookstep algorithm, we converge stable and unstable travelling wave solutions to the governing equations. By combining these two techniques, complex nonlinear pathways between attracting states are revealed, providing insight into a potential subcritical origin of the geodynamo. This paper showcases these methods on the widely studied benchmark of Christensen et al. (2001), laying the foundations for future studies in more extreme and realistic parameter regimes. We show that the minimal seed reaches a nonlinear dynamo solution by first approaching an unstable travelling wave solution, which acts as an edge state separating a hydrodynamic solution from a magnetohydrodynamic one. Furthermore, by carefully examining the choice of cost functional, we establish a robust optimisation procedure that can systematically locate dynamo solutions on short time horizons with no prior knowledge of its structure.

physics.flu-dyn↗

arXiv2025

Fast automated adjoints for spectral PDE solvers

We present a general and automated approach for computing model gradients for PDE solvers built on sparse spectral methods, and implement this capability in the widely used open-source Dedalus framework. We apply reverse-mode automatic differentiation to symbolic graph representations of PDEs, efficiently constructing adjoint solvers that retain the speed and memory efficiency of this important class of modern numerical methods. This approach enables users to compute gradients and perform optimization for a wide range of time-dependent and nonlinear systems without writing additional code. The framework supports a broad class of equations, geometries, and boundary conditions, and runs efficiently in parallel using MPI. We demonstrate the flexibility and capabilities of this system using canonical problems from the literature, showing both strong performance and practical utility for a wide variety of inverse problems. By integrating automatic adjoints into a flexible high-level solver, our approach enables researchers to perform gradient-based optimization and sensitivity analyses in spectral simulations with ease and efficiency.

math.NA↗

arXiv2024

Weakly nonlinear analysis of the onset of convection in rotating spherical shells

A weakly nonlinear study is numerically conducted to determine the behaviour near the onset of convection in rotating spherical shells. The mathematical and numerical procedure is described in generality, with the results presented for an Earth-like radius ratio. Through the weakly nonlinear analysis a Stuart--Landau equation is obtained for the amplitude of the convective instability, valid in the vicinity of its onset. Using this amplitude equation we derive a reduced order model for the saturation of the instability via nonlinear effects and can completely describe the resultant limit cycle without the need to solve initial value problems. In particular the weakly nonlinear analysis is able to determine whether convection onsets as a supercritical or subcritical Hopf bifurcation through solving only linear 2D problems, specifically one eigenvalue and two linear boundary value problems. Using this, we efficiently determine that convection can onset subcritically in a spherical shell for a range of Prandtl numbers if the shell is heated internally, confirming previous predictions. Furthermore, by examining the weakly nonlinear coefficients we show that it is the strong zonal flow created through Reynolds and thermal stresses that determines whether convection is supercritical or subcritical.

physics.flu-dyn↗

arXiv2020

A parallel-in-time approach for accelerating direct-adjoint studies

Parallel-in-time methods are developed to accelerate the direct-adjoint looping procedure. Particularly, we utilize the Paraexp algorithm, previously developed to integrate equations forward in time, to accelerate the direct-adjoint looping that arises from gradient-based optimization. We consider both linear and non-linear governing equations and exploit the linear, time-varying nature of the adjoint equations. Gains in efficiency are seen across all cases, showing that a Paraexp based parallel-in-time approach is feasible for the acceleration of direct-adjoint studies. This signifies a possible approach to further increase the run-time performance for optimization studies that either cannot be parallelized in space or are at their limit of efficiency gains for a parallel-in-space approach.

math.OC↗