Numerical approaches to the geodynamo problem

Earth’s magnetic field is generated and sustained in Earth’s core by the nonlinear dynamical process known as dynamo action. The core magnetic field penetrates Earth’s mantle and is observed and recorded on the surface of Earth. Scientific studies of the time evolution of Earth’s magnetic field follow two lines: (1) geodynamo modeling and simulation and (2) inversion of observed geomagnetic data. Following the approach of geodynamo modeling, namely solving the Navier-Stokes, the magnetic induction and the temperature equations simultaneously, we explored the possibilities for speeding up geodynamo computations in order to study a broader regime of control parameters. Spherical transforms are operations that transform from spectral to physical space and vice versa, and these spectral transforms of spherical harmonics are needed in every modern geodynamo code for the computation of nonlinear terms such as the advection of momentum. We tested the numerical complexity and accuracy of a fast algorithm for computing the spectral transform of spherical harmonics, invented by Driscoll and Healy (1994) and improved by Kunis and Potts (2003). Due to the stabilization step introduced by Kunis and Potts (2003), the fast algorithm exhibits cL3 complexity, instead of cL2 ln L complexity, where L is the maximum spherical harmonic degree, equivalent to the resolution of the model and c is a prefactor. The fast algorithm is 50–100 times slower than the conventional one for spherical harmonic degrees L ≤ 511. For the geodynamo modeling, we also found an optimal Galerkin scheme for solving the dynamo system in a full sphere and tested the numerical convergence of the radial basis functions against a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos (Bachtiar et al., 2006; Dudley and James, 1989; Gubbins et al., 2000). The radial basis functions are constructed from a terse sum of onesided Jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior, but is everywhere infinitely differentiable, including at the origin. We tested two types of orthogonal radial basis sets with respect to different weight functions, (1) Legendre type with unit integration weight, (2) Chebyshev type with weight function, w = 1/ √ 1− r2, and found that both radial basis sets exhibit exponential convergence, superior for a given problem size to any other scheme hitherto reported. Numerically, the Legendre type basis functions converges even faster than the Chebyshev type, therefore they are preferred for the numerical modeling of full sphere problems. Following the geomagnetic inversion approach, we developed a mathematical methodology, termed the adjoint dynamo model in continuous form, along with the corresponding numerical algorithm to apply to the problem of retrieving the spatial distribution of the velocity, magnetic and temperature fields in the past and predicting the time evolution of the fields in future by linking the surface magnetic observations with the underlying geodynamo model. Such an approach is termed variational data assimilation or 4DVar. We tested our analysis and the corresponding numerical algorithm on two illustrative problems in a whole sphere: one is a linear kinematic dynamo model, and the second is associated with the nonlinear Hall-effect dynamo. The algorithm exhibits reliable numerical accuracy and stability. Using closed-loop simulations with noise free data, we demonstrated the ability for the adjoint kinematic dynamo system and the adjoint Hall-effect system to retrieve unknown initial conditions, especially the ability to retrieve the unobserved toroidal magnetic field. Using both the analytical and numerical techniques we developed, the adjoint dynamo system can be solved directly with the same order of computational complexity as that required to solve the forward problem. The methodology now awaits application to real geomagnetic data, such as the global database of observations spanning the last 400 years (Jonkers et al., 2003).