10/

I wrote about this more on my 🌐website🌐 here:

borisandrews.github.io/publications/parker/

You can also check out my previous work, laying down a framework for designing such 🖥️conservative simulations🖥️ here:

borisandrews.github.io/publications/sp-integrators/

9/

TL;DR:

🅰️ If you're doing simulations of MAGNETIC RELAXATION, and you don't want your solutions just to vanish, you've got to conserve H...
🅱️ ...And if you want to conserve H, you should read our paper! :)

8/

This ensures COMPUTED MAGNETIC FIELD WON'T ARTIFICIALLY VANISH.
(You can see this in the first image from this thread!)

This is (essentially) the 🥇first simulation where the solutions don't just vanish🥇.

7/

In our work, we construct a numerical scheme that CONSERVES H EXACTLY👌.

You can see this in the figure below.
A typical simulation (dashed blue) has H artificially decay to 0; our simulation (solid red) holds H steady, and so 💪holds up E💪 too!

6/

Anyway, while the equations satisfy these laws, they're not necessarily preserved in any old simulation.

❌ NOT ALL SIMULATIONS ARE CREATED EQUAL ❌

In particular, EXISTING NUMERICAL SCHEMES TYPICALLY DO NOT CONSERVE H.

5/

H quantifies how KNOTTED the initial magnetic field is. Its conservation implies magnetic relaxation CANNOT UNTIE THESE KNOTS.
It loosens them (E decreases) but it never unties them.

</🪢 FUN TOPOLOGICAL DIVERSION 🪢>

4/

<🪢 FUN TOPOLOGICAL DIVERSION 🪢>

This has a neat topological interpretation!
(And we all love topology, right?)

H can be interpreted as a continuous version of the LINKING NUMBER. This represents the no. of times a pair of loops wind around each other (1, 2, 3 in the image below).

5/

A simple inequality (the ARNOLD INEQUALITY) says (kind of...) that:

"E can't pass below H".

With H constant, E can never reach 0; H literally💪holds up E💪. In the equilibrium state, the MAGNETIC FIELD SHOULD NOT VANISH.

4/

If E ever hits 0, the magnetic field has simply vanished everywhere.
The interest thing however is that...

❌ THIS SHOULD NEVER BE THE CASE ❌

3/

In this work, we develop accurate numerical simulations for a certain magnetic relaxation model:
the MAGNETO-FRICTIONAL EQUATIONS

🧲🧲🧲

These equations conserve a quantity called the helicity (H) and dissipate a quantity called the energy (E).

2/

MAGNETIC RELAXATION =
the process by which a magnetic or magnetohydrodynamic (MHD) system converges to equilibrium.

These systems are typically long-duration, large-scale plasmas (e.g. the Sun ☀️ in particular its corona visible in the photo below) or liquid metals (e.g. the Earth’s core 🌍).