Field lines, advection vs diffusion, and “frozen-in” behavior

11:13day that magnetic field is actually going parallel with the velocity of those particles because there's two cross productducts. This is something that people with um vector calculus knowledge would know, right? If you have two cross products, then actually you're just going in in the same direction. Um and then the second term there new over mu not >> del^ squ b that looks like a heat equation. Okay? And that is your diffusion term. Basically, you know, if you have a hot if you have a metal rod and you heat up one part of the metal rod, the heat is going to diffuse >> across >> across that metal rod, right? And the magnetic field in these plasmas does the same thing. If you've got a lot of magnetic field here, it's going to sort

11:53of diffuse and the magnetic field lines are going to diffuse out. >> So the the the propagation of the magnetic field kind of looks similar to the propagation of heat >> of heat >> from a mathematical perspective. >> From a mathematical perspective, but there's those two terms, right? There's the diffusion term, which is that heat kind of term. Yes. And then there's the um advection term which is the magnetic field moving in line with the charged particles because you've got that double dot um cross productduct. So when a plasma is a perfect conductor in that equation there was this new right which is the conductivity. >> Okay >> if [clears throat] if a plasma is perfect then the the the conductivity is

12:35just like amazing. There's no resistance. Okay. And in that case, we don't have to worry about that that second term, >> the the the the heat looking thing. >> Yeah. The heat looking thing is because it's just gonna like go >> go. Yeah. Yeah. >> Right. So, so we don't have to worry about that. And then we get something called the flow frozen influx theorem. Basically, what this means is that the magnetic field lines don't actually like diffuse out. They get locked in place in with these plasma particles. So on the left you see straight lines and then as the plasma particles move around the magnetic field lines move around with them. Got it? >> You know and the the field lines don't break. They don't cross. They don't like

13:15change their connectivity and in in some sense like topologically they're they're preserving this mathematics across those scales. Got it. >> Okay. So that would be really great. >> Yes. >> If that's what plasma did, >> right? And we didn't have to worry about one of the terms in the equation. It's still difficult, but you know, at least there's >> it's incrementally less difficult. >> Yeah, it's [laughter] incrementally less difficult, right? Like at least the the flux is frozen into the plasma, right? >> Okay. So that that's the that's that frozen in flux theorem. Yeah. >> As a means by which to try to uh uh calculate and understand magneto hydrodnamics. >> Yes. Yes. It would be like like a great approximation

13:56if we could do it. not dissimilar to the approximation we do with the the Stokes equation. Yeah. Like like in terms of we can't really model Exactly. >> Exactly. And so we need some kind of approximation. >> Yes. Yeah. So, so maybe maybe at like certain geometries we can like you know and obviously it works right like uh Formula 1 teams use the Navier Stokes equation to figure out >> how their car is going to do on a certain track right so obviously it works but then you get to really high