## Streaming Instability? Sup Brah!

Discuss how polywell fusion works; share theoretical questions and answers.

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Robthebob
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Location: Auburn, Alabama

### Streaming Instability? Sup Brah!

Just learned this in class yesterday, dont ask me to derive the math, I cant say I understand it 100%

Consider a function g0(v), v=velocity in a direction

g0(v)=f hat e0 + sum( (me/ms) f hat s0

f hat s0 is the zeroth order normalized distribution function of species s. So this is talking about at a position, for a velocity in a direction, there's a distribution of the particle of a species s. Like say at (x,y,z)=(x0,y0,z0), there are like a bunch of electrons, those electrons can be distributed based on their velocity in the x direction (there can different distribution in y and z). e is just electron. ms is the mass of a particle of species s.

turns out, if this function g0(v) has peaks, then it must have at least 1 valley, if any single valleys is deep enough, then it's unstable. Now what do I mean by deep enough?

Well... you asked... Again, dont ask me about the math, I can potentially get yall some stuff to read.

In order for a system to be two stream unstable, the following must be true.

P * Int (from -infinity to infinity) ((dg0/dv)/(v-x0)) dv >= 0

P is the Cauchy principle value (what is this? I actually dont know... need to go ask my boss), what is x0? I think this is location of a valley, I could be wrong, will go ask someone about this.

So in order to know if something is unstable (this doesnt answer if this instability will actually be a problem), is you have to do this analysis for every position in your system for all 3 velocities. In a polywell this matters only on magnetic fields lines and in the core of the machine.

It's very likely that this analysis will only matter inside the core, because most of the electrons and ions are inside the machine. This is also pretty darn hard to do, because you need to know f(x,y,z,vx,vy,vz)s0 for the polywell (we're assuming steady state, t doesnt cause things to vary), and that's pretty hard to do. You need a PIC code to even start looking at this.

Hopefully there's not going to be anymore nonsense about streaming instability here. Unless you did some heavy duty computation analysis, theoretical analysis, I (and probably everyone else) dont want to hear it. Joe.
Throwing my life away for this whole Fusion mess.

asdfuogh
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Location: California

### Re: Streaming Instability? Sup Brah!

Principal value integral is just like.. taking the main part of the integral such that your function is single valued, or something similar to that. There's probably more complicated or advanced catches and tidbits about it though.

Why is it that g0(v) must have a valley if it has a peak?

I'd love to start trying to do some very cruddy simulations to both contribute and practice my coding.. Unfortunately, I'm not even at that point yet. Also, I'm even not really sure how to pick coordinate systems to work with since a Polywell has open field lines..

Robthebob
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Location: Auburn, Alabama

### Re: Streaming Instability? Sup Brah!

i think i know what principle values mean, but i dont know how it applies in the context.

I said if a function have peaks (more than 1), it must have at least 1 valley. I imagine having to brute force it, which may be completely impossible. Rick Nebel wrote a paper on it, and probably already worked it out, but no one knows how that paper turned out. I was told that because the input electron is like nothing, it shouldnt matter. But then, there are different flows in electrons and ions in polywell, which I think unless you did particle in cell codes, you dont know how these flows work, I dont think it's possible to figure it out by thinking or deriving it.

I also dont know how fast these suckers develop, remember the electrons are always getting lost and replenished, which I dont know if that matters...
Throwing my life away for this whole Fusion mess.

kcdodd
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### Re: Streaming Instability? Sup Brah!

Maybe you could setup a simple numerical model. One radial dimension, and one radial velocity, thus only 2-D in phase space. You would setup the outer boundary so that the particles are simply reflected back: f(Rmax, -v) = f(Rmax, +v). You would assume some the background field with negative potential in the middle, then let it fly. Wouldn't account for other effects of magnetic field, and streaming through cusps etc.
Carter

asdfuogh
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Location: California

### Re: Streaming Instability? Sup Brah!

What are the normal coordinate systems used for numerical simulations in previous codes? Or, actually, what kind of coordinate system is usually used for a magnetic mirror system?

Robthebob
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Location: Auburn, Alabama

### Re: Streaming Instability? Sup Brah!

I dont know the usefulness of doing that, because the streaming stability of 1 path isnt going to be too valuable, because there are a huge number of paths for particles to take, unless this is the only 1 path that poses a problem, everything else is probably stable, which you dont know that.

My post is really to explain how difficult it is to even start answering this question, it's basically impossible to get much out of the simulation without a PIC code.

(I just remembered that renewing particles wont stop this instability from growing)
Throwing my life away for this whole Fusion mess.

kcdodd
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### Re: Streaming Instability? Sup Brah!

Yes it would have to be a pic code. But using spherical symmetry makes it simpler, and needs fewer particles, than if you attempted to use all 3 dimensions.
Carter

D Tibbets
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### Re: Streaming Instability? Sup Brah!

There are papers from EMC2 describing such a 2-D ( I thin k they refer to it as 1-D) from the early 90's. Look at

A couple of papers-

In some of them they describe the math with motions primarily confined to radial directions. This is defended by assumptions that most collisions occur in the core so that not much angular momentum (transverse motion) is introduced per average collision. With recognized cautions they used this to model much of the action in the Polywell. I don't know how useful these models were for predicting experimental results, though I have not heard of them being discredited.

Dan Tibbets
To error is human... and I'm very human.

Robthebob
Posts: 383
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Location: Auburn, Alabama

### Re: Streaming Instability? Sup Brah!

kcdodd wrote:Yes it would have to be a pic code. But using spherical symmetry makes it simpler, and needs fewer particles, than if you attempted to use all 3 dimensions.
I'm not completely convinced, the machine is somewhat symmetric. I personally dont know how where the guns are is gonna affect the symmetry. The two guns are from two opposing faces... we can probably assume that every slice from 1 face to another about the very center would be symmetrical.

Slices running up and down and back and forth should be symmetrical. So you only need to sim a quarter of it... I dont know if you can say within that quarter there are anymore place you can claim symmetry.
Throwing my life away for this whole Fusion mess.

kcdodd
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### Re: Streaming Instability? Sup Brah!

Asymmetry might alter the exact answer, but when we don't even know if there is an instability, or if there is what the growth rate is or how it manifests itself in scale or frequency, a very simple model seems like the best first step. Then if you chose to make it more complicated you have already worked out any issues that might exist with particual configuration, and you have already approximate solutions to compare the more complex model against. I have found that starting super simple is valuable, even if it may not be exactly the thing you're intending to end up with.
Carter

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### Re: Streaming Instability? Sup Brah!

I have to agree with Carter here.

Start simple, and then grow the complexity of your sim as you learn from it.

One of the issues for the modelled zone can be resolved if you assume fully symmetrical gun placement for the full device.
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KitemanSA
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### Re: Streaming Instability? Sup Brah!

My recollection is the Dr. B worked with what he called a 1 1/2 D code, all radial values plus angular momentum IIRC.