Hitchhiker's guide to the Outer Core: The Induction Equation

This article is a second installment in my series on the Geodynamo and our Three Meter laboratory experiment that models it. To review, the earth's outer core is home to an important process which scientists refer to as the Geodynamo (Geo = Earth, dynamo = power; thus it describes how the magnetic field of our Earth is powered). This process is responsible for the generation of magnetic fields in the liquid metal of the outer core. 

We continue our story on the Geodynamo by looking at the important governing equation: the Induction Equation. As its name suggests, this mathematical model describes how the magnetic field is induced, or created, from the dynamics of the fluid flow. We quantitatively describe the flow of a fluid by its velocity field. For a typical laminar (or smooth) flow, it is fairly simple and predictable. However, as we increase the size and the velocity of our system, we have a more complex turbulent flow. 

Velocity field lines for laminar flow (left) and turbulent flow (right) for fluid through a cross-section of a pipe. Laminar flow lines are smooth, straight, and follow a straightforward pattern. The turbulent scenario is messier; there are swirls, or eddies, in the flow. Source: https://www.michael-smith-engineers.co.uk/resources/useful-info/pipe-velocity

The other major part of the induction of the magnetic field comes from the electromagnetic properties of the fluid. As such, we use Maxwell’s equations for changing electric and magnetic fields, and Ohm's law relating current density to the electric field:

Combining these equations, we can cancel out E and J and arrive at the Induction equation:

This is very similar to the equation for the vorticity (ω) of a fluid, or how much the fluid swirls, and is based on the curl of the conservation of momentum for a fluid:

This analogy to the vorticity equation is rather profound. The first term is a common diffusion term, showing that energy is lost by the magnetic diffusivity (η), which is associated with the electrical resistance of the material, like heat dissipating through friction. The second term is analogous to vortex stretching: imagine you are holding a string vertically on both ends, and in the middle a ball is attached. As you swing the ball around, you sweep a circle roughly parallel with the ground. But pull the two ends together and the ball gets faster. The general idea is reflected as vortices, or swirls in the fluid, naturally elongate, or stretch, as a property of turbulence. To not overcomplicate things, the general idea is that as these vortices stretch the preexisting magnetic field, thereby strengthening it. 

A rotating part of the fluid will tend to stretch out perpendicular to the plane of rotation resulting in smaller eddies spinning faster. This transition from energy held in large vortices to smaller vortices is due to the turbulent energy cascade. The top right of the video briefly shows this concept. Source: https://www.youtube.com/watch?v=_UoTTq651dE&pp=ygUONS8zIHR1cmJ1bGVuY2U%3D
 

The Induction equation is inherently complex and nonlinear due to this stretching term; a steady increase in the velocity of the fluid does not mean a steady increase in the magnetic field strength. This is because magnetic fields interact with the fluid’s electrical conductivity. As the fluid moves, it generates electric currents, altering the magnetic field. Just as vorticity amplifies due to fluid interactions, the magnetic field strength responds nonlinearly to fluid motion and electromagnetic effects.

On account of the equation's nonlinearity, solving it analytically is difficult. Computational simulations use advanced numerical methods and algorithms that break down the equation into manageable steps and approximate the solutions over small time intervals. Over the past 30 years, simulations based on the Induction equation have been successful in replicating a dynamo, but it is still unclear as to what the exact dynamics are, specifically on earth, that have allowed this magnetic amplification. 

A snapshot of the simulated geomagnetic field produced by Glatzmaier and Roberts (1995). The lines follow the paths of magnetic fields generated from fluid motions. This is taken during a supposed polar reversal.

The core reason numerical simulations cannot model earth's dynamics ultimately stems from the concept of energy cascades, via turbulence. Large eddies literally cascade into smaller eddies; you may notice this when you stir milk into coffee and watch what happens over time. To model both the large eddies in earth's outer core down to the teeniest of eddies on the micrometer size, you need extremely high resolution and computing power, that which even the best supercomputers cannot achieve.

Smoke and air mixing over time, illuminated by a green planar laser. Large eddies are visible but these all break down into smaller and smaller eddies. Source: https://www.youtube.com/watch?v=_UoTTq651dE&pp=ygUONS8zIHR1cmJ1bGVuY2U%3D

References

1. R. Beck, Magnetic fields in spiral galaxies, The Astronomy and Astrophysics Review, 24, 4 (2015)

2. Bondi, Hermann Sir and Thomas Gold. “On the Generation of Magnetism by Fluid Motion.” Monthly Notices of the Royal Astronomical Society 110 (1950): 607-611.

3. Glatzmaiers, G., Roberts, P. A three-dimensional self-consistent computer simulation of a geomagnetic field reversal. Nature 377, 203–209 (1995). https://doi.org/10.1038/377203a0