Jitse Niesen

The paper "The inherent instability of axisymmetric magnetostrophic dynamo models", written by Colin Hardy, Phil Livermore and myself, is now published online in the journal Geophysical & Astrophysical Fluid Dynamics.

Link (open access): https://www.tandfonline.com/doi/full/10.1080/03091929.2022.2148666

Below I will give a (relatively) non-technical explanation of some background of our paper.

#geomagnetism #geodynamo #EarthsCore #magnetohydrodynamics #magnetostrophic

The inherent instability of axisymmetric magnetostrophic dynamo models

Recent studies have demonstrated the possibility of constructing magnetostrophic dynamo models, which describe the slowly evolving background state of Earth's magnetic field when inertia and viscos...

Taylor & Francis
Dec 3, 2022 at 1:52pmWeb
Show threadJitse NiesenDec 3, 2022

The Earth's magnetic field originates in the outer core of the Earth, which consists of molten metal. The metal is not stationary but flows and this generates the magnetic field via the dynamo effect.

Here is the obligatory XKCD cartoon.

Show threadJitse NiesenDec 3, 2022
Mathematically, the Navier-Stokes equation describe how a fluid, such as the molten metal in the core, flows and Maxwell's equations describe how the electric and magnetic fields evolve. Magnetohydrodynamics (MHD) is the study of these equations together. People solve the MHD equations numerically to try to understand the details of the process.
Show threadJitse NiesenDec 3, 2022

The Navier-Stokes equation contains six terms:

1. The Lorenz force which the magnetic field exerts on the metal.
2. Gravity, which causes heavier metal (e.g., colder metal) to move down.
3. Pressure, an outward force like water or air pressure.
4. Viscosity, a friction when adjacent layers of metal flow with different velocities and slide against each other.
5. The Coriolis force, which arises because the Earth rotates.
6. The five forces above equal mass times acceleration by Newton's law.

Show threadJitse NiesenDec 3, 2022
Howvever, the six terms are not equally important. When looking at the large-scale movement in the Earth's core, the viscosity and acceleration (terms 4 and 6) are small compared to the other terms. This is quantified by the so-called Ekman number and Rossby number, which are approximately \( 10^{-15} \) and \( 10^{-9} \), respectively.
Show threadJitse NiesenDec 3, 2022

These very small numbers make it extremely difficult to do computer simulations, so simulations of the whole core have to be run with much greater values of the Ekman and Rossby number.

In our paper, we take a different approach: we ignore the viscosity and acceleration and assume that the four remaining terms balance. This is called the magnetostrophic balance and amounts to setting the Ekman and Rossby number equal to zero.

Show threadJitse NiesenDec 3, 2022
Our approach is not new and seems quite natural, but it is not that popular. The reason is that even though it is quantitatively a small change, it completely changes the character of the equation. In maths speak, it is a singular perturbation.
Show threadJitse NiesenDec 3, 2022
If you want to know more, you need to look at the paper. In short, we need some more assumptions: the system is axisymmetric (which we know cannot be the case), there is no inner core (also not true), and the buoyancy force has a specific form. We then compute steady states of the system and show the system typically does not reach these. This means that the magnetic field varies all the time, which is the "inherent instability" in the title.
Show threadJitse NiesenDec 3, 2022
PS: I am well aware that there are people here that know the field far better than me, and I hope they will correct or clarify where my explanation is lacking.
Show threadAbbasBalloutDec 3, 2022
@jitseniesen i'd like a technical summary. Especially the numerics involved.
Show threadJitse NiesenDec 4, 2022
@AbbasBallout For technical details, read the paper :) But not much numerics. We use a fairly complicated spectral discretization in space and Maple to find stationary points. For the hard part (time stepping), we use the Fortran code by Kuan Li et al. described in https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/taylor-state-dynamos-found-by-optimal-control-axisymmetric-examples/C6977FA6E9A93B23900E39BCB1B9FEC4
Taylor state dynamos found by optimal control: axisymmetric examples | Journal of Fluid Mechanics | Cambridge Core

Taylor state dynamos found by optimal control: axisymmetric examples - Volume 853

Cambridge Core
Show threadJitse NiesenDec 4, 2022
@AbbasBallout Everything is done in spherical coordinates to match the boundary conditions. We use a Galerkin basis consisting of spherical harmonics, and linear combination of Jacobi polynomials in the radial direction. We use that both magnetic field and fluid velocity are div free. Almost the same basis is used for both, but the linear combination is slightly different to match the boundary conditions. IIRC we use about 60 modes in each dimension, so 60^3 in total.