From the Auroral Electrojet to the Deep Ocean

Since my last update, I’ve been working almost entirely on a nasty numerical problem at the slow end of the magnetic field spectrum. The basic electromagnetic system is still:

A(ω) = K − iωMσ

At very low frequencies, part of that system becomes very badly conditioned. So, pulled the unstable part out and started solving it separately. That reduced problem is now:

H_red = C⁻¹ + VA₀⁻¹U

The first version got really close, but not close enough:

Prototype 1 true residual ≈ 1.24 × 10⁻⁴

My gate is:

Required reduced residual ≤ 1 × 10⁻⁶

So, I tried a diagonal preconditioner, and it made things so much worse. :(

Prototype 2 true residual ≈ 2.21 × 10⁻¹

It told me the different spatial modes are strongly coupled, so I’m now testing whether I can split them into low-, middle-, and high-degree blocks and solve them in a better way. And, this connects to something Ben Davidson pointed out to me.. Ben’s observation was that during lower level geomagnetic activity, the strongest induction related activity is concentrated much farther north, around the auroral electrojets and polar regions. As storms become stronger, that active region can push farther toward lower latitudes.

Right now, my external magnetic forcing is still too spatially coarse to properly resolve a localized auroral electrojet or polar cusp structure, but now I’m building toward it.

Once I get this low frequency numerical problem qualified, the next forcing upgrade is going to include much more spatial structure. Then I can ask whether the modeled currents and Joule heating really stay concentrated at high latitudes during weaker activity and expand equatorward as the forcing gets stronger.

I want this solver to tell me where the electromagnetic energy goes.

And eventually, how much of it reaches the deep ocean.