Streaming Instability? Sup Brah!
Posted: Fri May 03, 2013 1:23 am
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.
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.