I’m so excited about this update! Since my last post, I’ve spent a lot of time working through the numerical problems in my electromagnetic induction solver. We finally have a major breakthrough. The reduced system I’ve been trying to solve is:
The biggest problem was memory. Every new calculation was taking up more space, and eventually the solver would run out. I also had some incredibly small numerical errors that were stopping otherwise accurate calculations. After fixing the memory management and refining the numerical acceptance tests, I finally completed the entire reduced operator!
This means I finally finished building the mathematical model that describes how electricity can be induced inside Earth and its oceans! Before this, I was struggling with a massive 35,640 x 35,640 matrix that wouldn't converge using Krylov methods. Every correction seemed to cause another issue, and the whole thing started spiraling into chaos.
But I finally completed the entire matrix, built a new direct solver, and now we're actually running the calculation! If it passes numerical validation, I can start reconstructing the induced electric fields and currents to find out how much heat they could generate in the deep ocean.
This is a huge milestone, guys, so huge. We now have the complete mathematical system needed for the next stage. I’ve built and locally tested a direct numerical solver, and we’re preparing to solve:
Once that calculation is successful and independently verified, we can begin reconstructing the induced electric fields and electrical currents inside Earth. So how close are we to answering the ocean heating question? The quantity I want to calculate is Joule heating:
That gives the rate of Joule heating per unit volume. Then we can calculate the heating throughout the deep ocean and compare it with the warming signal we’ve been investigating.
Questions to answer with the EM Solver
Ok, so first is going to absolutely be the abyssal ocean heating question, but after that… so many questions we can answer. My favorite so far comes from Tim @polewanderer who wanted to know if electrical currents induced inside Earth generate enough heat to change the viscosity of the asthenosphere.
The asthenosphere is mostly solid rock that flows very slowly, with some regions containing small amounts of partial melt. Its viscosity is extremely sensitive to temperature, and even slight heating can change how easily the rock deforms.
We already know that changing magnetic fields can induce electrical currents inside Earth. Those currents generate heat through electrical resistance, called Joule heating.
So… could planetary electrical currents generate enough heat to reduce mantle viscosity?
We can calculate the induced currents, estimate the resulting temperature changes, and use established mantle rheology equations to predict how viscosity responds. I also want to investigate whether electric fields could directly influence the behavior of partially molten rock.
The exciting part is that we can quantify what would actually be required. How strong would the currents need to be? How long would they need to persist? And do naturally occurring geomagnetic disturbances produce anything close.
