Towards millenial-timescale geodynamo models with zero viscosity

One of the great challenges facing geodynamo modellers is that, owing to limitations of current modelling techniques and computational resources, values of the accessible parameters are remote from those in the Earth's core. This means that making robust inferences from models about the geodynamo and associated processes in the core is extremely difficult, if not impossible. In a typical nondimensionalisation of the geodynamo equations, two key parameters that appear are very small: the Rossby number (measuring the strength of inertia) and the Ekman number (measuring viscosity). Since the Ekman number is O(10^-15) in the core, viscosity may be readily neglected. On timescales comparable to the magnetic diffusion time, on the order of millenia, the Rossby number is also small, being O(10^-7) or so and may also be neglected. On shorter timescales, it is known that inertial effects remain important, for instance, driving torsional oscillations. Dropping both inertia and viscosity from the governing equations results in the magnetostrophic balance, featuring only the Coriolis, pressure, buoyancy and Lorentz forces. However, at all times it is known that any associated magnetic field must necessarily satisfy Taylor's constraint (after Taylor, 1963), a condition which imposes a tight straitjacket on its structure. This constraint enforces as zero the azimuthal component of the Lorentz force (depending on the magnetic field) when averaged over geostrophic contours: fluid cylinders aligned with the rotation axis. The inherent technical difficulties associated with this condition, not least being the infinite number of contours over which the constraint must be satisfied, have hampered the development of inviscid models since the 1960's. Very recently, we discovered a general framework in which we could derive magnetic fields that exactly satisfied Taylor's constraint. These theoretical ideas are based on the representation of the magnetic field as an appropriately chosen truncated Galerkin expansion, under which Taylor's constraint collapses to a finite number of conditions, thus admitting the possibility of a numerical method that evolves the magnetic field along the so-called Taylor manifold, the multi-dimensional surface of magnetic field solutions satisfying the Taylor constraints. The development of such an algorithm is not straightforward however as, despite the number of constraints being finite, each is quadratic in the spectral magnetic field coefficients. Although the geostrophic flow (constant on geostrophic contours) formally confines the evolution of the field to the Taylor manifold, numerical error causes an inevitable exponentially growing departure and thus a more robust approach is required. In this presentation, we propose an alternative scheme with the magnetic field evolved by projecting its time rate of change onto the local tangent plane to the manifold. In an example, we demonstrate that a diffusing magnetic field may be evolved over many millenia while maintaining Taylor's constraint to very high accuracy.