A computationally efficient spectral method for modeling core dynamics

Abstract

An efficient, spectral numerical method is presented for solving problems in a spherical shell geometry that employs spherical harmonics in the angular dimensions and Chebyshev polynomials in the radial direction. We exploit the three‐term recurrence relation for Chebyshev polynomials that renders all matrices sparse in spectral space. This approach is significantly more efficient than the collocation approach and is generalizable to both the Galerkin and tau methodologies for enforcing boundary conditions. The sparsity of the matrices reduces the computational complexity of the linear solution of implicit‐explicit time stepping schemes to O(N) operations, compared to O(N2) operations for a collocation method. The method is illustrated by considering several example problems of important dynamical processes in the Earth's liquid outer core. Results are presented from both fully nonlinear, time‐dependent numerical simulations and eigenvalue problems arising from the investigation of the onset of convection and the inertial wave spectrum. We compare the explicit and implicit temporal discretization of the Coriolis force; the latter becomes computationally feasible given the sparsity of the differential operators. We find that implicit treatment of the Coriolis force allows for significantly larger time step sizes compared to explicit algorithms; for hydrodynamic and dynamo problems at an Ekman number of E=10−5, time step sizes can be increased by a factor of 3 to 16 times that of the explicit algorithm, depending on the order of the time stepping scheme. The implementation with explicit Coriolis force scales well to at least 2048 cores, while the implicit implementation scales to 512 cores.

Connections between nonrotating, slowly rotating, and rapidly rotating turbulent convection transport scalings

“…This system is gravitationally unstable and drives buoyant convective flows across the fluid layer that advect both heat and momentum. We describe this system generally throughout this paper, but it can be thought of as an extended plane layer [14], a finite cylinder [15], or a spherical shell of fluid [16].…”

pizza: an open-source pseudo-spectral code for spherical quasi-geostrophic convection

“…At this stage, any single term that enters the above equation can be written as the product x q ∂ p f /∂x p , where p and q are positive integers. Following Marti et al (2016), this equation is then integrated by parts until no differential operator remains, such that each term has the following form…”

“…The internal matrix elements are determined using the freely available python package developed by Marti et al (2016) ‡ that allows the symbolic computation of those operators § . Excluding boundary conditions, A I m , B I m and C I correspond to band matrices with pu super-diagonals and p sub-diagonals that have a bandwidth defined by…”

Modelling the Earth's geomagnetic environment on Cray machines using PETSc and SLEPc

“…We adopted the PETSc/SLEPc approach because PETSc is already installed on Cray machines (although the version we use in these experiments is a later version due to a number of important bug fixes that have not yet been merged into the Cray official release), fairly ubiquitous in HPC, known to work well and the flexibility provided by these libraries is important. SLEPc has also been demonstrated to work well in a variety of application areas including engineering, materials science, structural analysis, Earth sciences, and geomagnetism . This last use case is important because it is the same research area as the MEME model and as such SLEPc has already demonstrated benefit in the domain of geomagnetism.…”