The consequences of quasineutrality in the cusps

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The consequences of quasineutrality in the cusps
In another thread I presented a calculation (again), showing that the plasma in the cusp (not only in the main plasma ball) must be quasineutral. I seem to have most everybody on board on that point. Now I'd like to examine the consequences of that fact.
Radially outward from the cusps, the magnetic field lines go more or less straight to the walls. The field will drop to some extent, so the mirror effect will tend to push the plasma outward, but we can ignore that for now. More important is the potential drop from the cusps to the wall. This drop should be large enough to turn back almost all of the electrons. For the ions, it's a downhill ride.
Qualitatively, if the ions have a small speed (fluid or thermal) in the cusp, the ones near the outside will drop off into the deep, leading to a smaller density on the outside and consequently an electric field that tends to pull additional ions after them.
Physically and mathematically, the situation is very similar to the Debye sheath. Bohm showed that the only stable solution is when the ions enter the region with the sound speed ( c_s = sqrt(kT_e/m_i) ). He assumed Maxwellian electrons and cold ions, two assumptions we may need to examine in more detail before we are through.
So my claim is that the ions, which have a density in the cusp only a small factor lower than the density in the central ball, will also be moving at a speed near c_s. They will smash into the wall with an energy on the order of the magrid potential. If this is all correct, the energy loss rate will be fatal.
P.S. Now that it sounds like Rick Nebel is not interested in grappling with the details of a polywell theory, this thread might be pretty lonely. Tom Ligon seems to still have some arrows in his quiver, so maybe I still have a chance of learning something here.
Radially outward from the cusps, the magnetic field lines go more or less straight to the walls. The field will drop to some extent, so the mirror effect will tend to push the plasma outward, but we can ignore that for now. More important is the potential drop from the cusps to the wall. This drop should be large enough to turn back almost all of the electrons. For the ions, it's a downhill ride.
Qualitatively, if the ions have a small speed (fluid or thermal) in the cusp, the ones near the outside will drop off into the deep, leading to a smaller density on the outside and consequently an electric field that tends to pull additional ions after them.
Physically and mathematically, the situation is very similar to the Debye sheath. Bohm showed that the only stable solution is when the ions enter the region with the sound speed ( c_s = sqrt(kT_e/m_i) ). He assumed Maxwellian electrons and cold ions, two assumptions we may need to examine in more detail before we are through.
So my claim is that the ions, which have a density in the cusp only a small factor lower than the density in the central ball, will also be moving at a speed near c_s. They will smash into the wall with an energy on the order of the magrid potential. If this is all correct, the energy loss rate will be fatal.
P.S. Now that it sounds like Rick Nebel is not interested in grappling with the details of a polywell theory, this thread might be pretty lonely. Tom Ligon seems to still have some arrows in his quiver, so maybe I still have a chance of learning something here.
Art,
I was under the impression that the cusp losses represented a vector problem. If the plasma is circulating and strongly directional wouldn't that influence the loss rate?
Perhaps the wiffle ball is not so much a closing of the holes as it is of the plasma taking a direction that is not in the loss cone.
But you know my thing is control and communications so I may be all wet here.
I was under the impression that the cusp losses represented a vector problem. If the plasma is circulating and strongly directional wouldn't that influence the loss rate?
Perhaps the wiffle ball is not so much a closing of the holes as it is of the plasma taking a direction that is not in the loss cone.
But you know my thing is control and communications so I may be all wet here.
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MSimon wrote:I was under the impression that the cusp losses represented a vector problem. If the plasma is circulating and strongly directional wouldn't that influence the loss rate?
Perhaps the wiffle ball is not so much a closing of the holes as it is of the plasma taking a direction that is not in the loss cone.
I guess. Write some equations and I'll try to poke holes in them.

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MSimon wrote:I was under the impression that the cusp losses represented a vector problem. If the plasma is circulating and strongly directional wouldn't that influence the loss rate?
Perhaps the wiffle ball is not so much a closing of the holes as it is of the plasma taking a direction that is not in the loss cone.
The magnets are a source of angular momentum. It either cancels out or the plasma has to spin.
And then there's surely turbulence...
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Re: The consequences of quasineutrality in the cusps
Art Carlson wrote:In another thread
P.S. Now that it sounds like Rick Nebel is not interested in grappling with the details of a polywell theory, this thread might be pretty lonely. Tom Ligon seems to still have some arrows in his quiver, so maybe I still have a chance of learning something here.
I don't think that Rick is not interested, as much as constrained by the Navy gag order.
I think we agree with you on quasineutrality in the cusps. As to the applicability of the Bohm theory, the ions should be cold since the average ion temperature should be lower than the central potential by a handful of times. Thus the ions flowing through the point of maximum potential will be the energetic tail of the Maxwellian, shifted down in velocity by the potential energy of the well. They probably won't themselves be Maxwellian, for what it's worth. The electron distribution will be pretty skewed in phase space, with the parallel velocity (relative to Bfield) much greater than the perpendicular, and, if the lifetimes are low as Dr. Nebel suggests, not really Maxwellian.
At the actual cusp throat, we know there will be a saddle point in the potential distribution. Beyond this point, where the potential decreases, the density of ions is going to be lower, and the phase space distribution is going to be lopsided, since few ionswill have velocity inward through the cusp. The lopsidedness will grow until all the ions at a given point are moving outwards. At that point, the average ion velocity times the density integrated over the area in each cusp should give the loss rate of ions in # particles/second. That seems like a reasonable calculation to make. We can make a pretty good guess at the area already, and the density.
Here's another handy drawing:
I have some misgivings about it though, and I will be disappointed if there is no debate.
One other way to treat this subject is as Dolan does, is to say that the probability of an ion being lost is some function that is zero up til approximately the point where the ions kinetic energy is equal to the well depth; at that point the probablity goes quickly to one as energy increases. It could be a simple step function. Under this condition, the distribution is more or less truncated, and the number of ions lost is proportional to the number that would ordinarily occupy that part of the distribution, dependent on the collision timescale. I've got the equations Dolan comes up with if you'd like me to type them up. This treatment could be applied to electrons too, which would tell us about the electron recycling rate and energy loss rate.
At the actual cusp throat, we know there will be a saddle point in the potential distribution. Beyond this point, where the potential decreases, the density of ions is going to be lower, and the phase space distribution is going to be lopsided, since few ionswill have velocity inward through the cusp. The lopsidedness will grow until all the ions at a given point are moving outwards. At that point, the average ion velocity times the density integrated over the area in each cusp should give the loss rate of ions in # particles/second. That seems like a reasonable calculation to make. We can make a pretty good guess at the area already, and the density.
Here's another handy drawing:
I have some misgivings about it though, and I will be disappointed if there is no debate.
One other way to treat this subject is as Dolan does, is to say that the probability of an ion being lost is some function that is zero up til approximately the point where the ions kinetic energy is equal to the well depth; at that point the probablity goes quickly to one as energy increases. It could be a simple step function. Under this condition, the distribution is more or less truncated, and the number of ions lost is proportional to the number that would ordinarily occupy that part of the distribution, dependent on the collision timescale. I've got the equations Dolan comes up with if you'd like me to type them up. This treatment could be applied to electrons too, which would tell us about the electron recycling rate and energy loss rate.

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 Cold ions at the edge: I think the ions will be hot and Maxwellian everywhere. I hope I don't need to make this assumption for present purposes though.
 Anisotropic electrons: I think the electrons will also be Maxwellian, but I also hope to do without this assumption.
 Saddle point in the cusp throat: I suspect this is also wrong. I think the potential along a cusp axis will drop monotonically from the center to the wall. If this turns out to be important, we will have to come back to it.
 "excess electron # density": Don't know what you mean by this. Otherwise your sketch looks like a good place to start.
As you say, the key is the ion velocity through the cusps. I am claiming that will be near the sound speed. Think of any gas streaming through a hole. Does anyone want to contest that? If not, then we can concentrate on the correct formula to use to calculate the sound speed.
Art said:
You are referring to choked flow from compressible flow theory I assume? For what reason will the ion flux be choked at that point?
Bit of a leap of faith to use a concept from compressible continuum fluid mech. for one species of a mixed particle, nonMaxwellian gas flow.
And then there's the electrodynamics to consider. If you are saying the ion has "vented" from a region of continuous plasma into a vacuum and is now an isolated particle having all its temperature as kinetic energy, then the ion kinetic energy must be now connected intimately with the field potential gradients in that region and the particles potential energy.
Yep, I guess you could say that I contest your assumption.
I am claiming that will be near the sound speed. Think of any gas streaming through a hole. Does anyone want to contest that?
You are referring to choked flow from compressible flow theory I assume? For what reason will the ion flux be choked at that point?
Bit of a leap of faith to use a concept from compressible continuum fluid mech. for one species of a mixed particle, nonMaxwellian gas flow.
And then there's the electrodynamics to consider. If you are saying the ion has "vented" from a region of continuous plasma into a vacuum and is now an isolated particle having all its temperature as kinetic energy, then the ion kinetic energy must be now connected intimately with the field potential gradients in that region and the particles potential energy.
Yep, I guess you could say that I contest your assumption.
cold ions at the edge: my contention is that the electric potential in the cusp will be such that by energy conservation the ions in the cusp throat will be lower energy. I'm imagining the ion behavior being more or less collisionless I suppose. By "edge" do you mean cusp, or the entire sheath in general?
electrons: if the electron confinement time is relatively short compared to the thermalization time, then the electrons will be mostly isotropic in the cusp, assuming that they are accelerated into the cusp by the potential of the well depth, and that the plasma potential is at ground. If the accelerating voltage of the well depth imparts enough electron energy relative to the electron temperature in the center, then the electrons will behave more like a beam with a certain energy spread.
saddle potential: this is a critical assumption for my previous points. It is an assumption. I'm open to rethinking it. In fact, it might be instructive to do so either way.
excess electron # density: I mean the electron density minus the ion density at a given point. The shape of the curve is an admitted WAG, moreso than the other curves.
The speed of sound requires electron temperature and ion mass only? That shouldn't be too tough. Electron temperature could be just the well depth (say 30keV), when the temperature is evaluated at the saddle point, ion mass is just deuterium I suppose. If I did my units right (convert eV to joules for numerator, use denominator in kg) then that's ~ 1.2e6 m/s. Dolan gets 1.4e4 m/s for his cusps. Not sure how.
I'll contest your speed of sound hypothesis.
As I recall, the speed of sound is derived for the case where the plasma expands in a vacuum so that the escaping electrons pull the ions behind them by electrostatic force. In this case the electrons are going both ways. I don't think the speed of sound necessarily applies like that here. The ion velocity we are looking for is the velocity of ions coming from the central plasma and having reached the saddle point in potential. If the central plasma is Maxwellian then ions are scattered into the range of total energy necessary to pass through the cusp. The higher the temperature of the central ions is, relative to the saddle potential, the faster the ions are upscattered and lost. The faster they are upscattered, the more likely they are to have a substantial amount of kinetic energy (velocity) left when passing through the cusp. Also, the longer it takes an energetic ion to reach the cusp, the more chance it will be upscattered higher.
electrons: if the electron confinement time is relatively short compared to the thermalization time, then the electrons will be mostly isotropic in the cusp, assuming that they are accelerated into the cusp by the potential of the well depth, and that the plasma potential is at ground. If the accelerating voltage of the well depth imparts enough electron energy relative to the electron temperature in the center, then the electrons will behave more like a beam with a certain energy spread.
saddle potential: this is a critical assumption for my previous points. It is an assumption. I'm open to rethinking it. In fact, it might be instructive to do so either way.
excess electron # density: I mean the electron density minus the ion density at a given point. The shape of the curve is an admitted WAG, moreso than the other curves.
The speed of sound requires electron temperature and ion mass only? That shouldn't be too tough. Electron temperature could be just the well depth (say 30keV), when the temperature is evaluated at the saddle point, ion mass is just deuterium I suppose. If I did my units right (convert eV to joules for numerator, use denominator in kg) then that's ~ 1.2e6 m/s. Dolan gets 1.4e4 m/s for his cusps. Not sure how.
I'll contest your speed of sound hypothesis.
As I recall, the speed of sound is derived for the case where the plasma expands in a vacuum so that the escaping electrons pull the ions behind them by electrostatic force. In this case the electrons are going both ways. I don't think the speed of sound necessarily applies like that here. The ion velocity we are looking for is the velocity of ions coming from the central plasma and having reached the saddle point in potential. If the central plasma is Maxwellian then ions are scattered into the range of total energy necessary to pass through the cusp. The higher the temperature of the central ions is, relative to the saddle potential, the faster the ions are upscattered and lost. The faster they are upscattered, the more likely they are to have a substantial amount of kinetic energy (velocity) left when passing through the cusp. Also, the longer it takes an energetic ion to reach the cusp, the more chance it will be upscattered higher.
Last edited by Solo on Sun Feb 08, 2009 8:03 pm, edited 2 times in total.

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So you want me to sing for my supper. I am willing to stick with cold ions, even though I believe they won't be cold. This will give a lower limit to the streaming velocity and the energy loss, and it also makes the theory easier.
The Bohm criterion says the ions will have a velocity equal to or greater than sqrt(kT_e/m_i), if the electrons have a Maxwellian distribution with temperature T_e. What we need to do is to generalize the Bohm criterion to a nonMaxwellian electron energy distribution. The general distribution would be an interesting problem, but also a bit of serious plasma theory that I probably don't have time for. I am working on a shortcut, but I don't know if it will work. What I can probably handle is a twotemperature electron distribution. Would that be an acceptable place to start?
The Bohm criterion says the ions will have a velocity equal to or greater than sqrt(kT_e/m_i), if the electrons have a Maxwellian distribution with temperature T_e. What we need to do is to generalize the Bohm criterion to a nonMaxwellian electron energy distribution. The general distribution would be an interesting problem, but also a bit of serious plasma theory that I probably don't have time for. I am working on a shortcut, but I don't know if it will work. What I can probably handle is a twotemperature electron distribution. Would that be an acceptable place to start?
Well, ok. There are likely to be both cold electrons oscillating in the cusp as well as fast electrons fresh from the electron injector.
I suppose this may show my ignorance, but I'm not sure what the electron speed has to do with the ion distribution. That's what I was getting at. I'll see about doing some more homework.
I suppose this may show my ignorance, but I'm not sure what the electron speed has to do with the ion distribution. That's what I was getting at. I'll see about doing some more homework.
Here's what we know and what we don't know:
1. We don't have the spatial resolution of the density to see if the cusps are quasineutral on the WB7
2. In oneD simulations the plasma edge (which corresponds to the cusp regions) is not quasineutral. Therefore, if the cusps are quasineutral it must be a multidimensional effect.
3. Energy confinement on the WB7 exceeds the classical predictions (wiffleball based on the electron gyroradius) by a large factor.
Our conclusion is that both the wiffleball and the cusp recycle are working at a reasonable level.
1. We don't have the spatial resolution of the density to see if the cusps are quasineutral on the WB7
2. In oneD simulations the plasma edge (which corresponds to the cusp regions) is not quasineutral. Therefore, if the cusps are quasineutral it must be a multidimensional effect.
3. Energy confinement on the WB7 exceeds the classical predictions (wiffleball based on the electron gyroradius) by a large factor.
Our conclusion is that both the wiffleball and the cusp recycle are working at a reasonable level.
Art:
Maybe.
Even so though, it looks to me like the Bohm integration doesn't hold because the boundary conditions of the plasma do not meet the Bohm sheath criterion bc's. There is a potential field due to Magrid outside the plasma to consider
http://en.wikipedia.org/wiki/Debye_sheath
"At the sheath edge (ξ = 0), we can define the potential to be zero (χ = 0) and assume that the electric field is also zero (χ' = 0)."
If so, the potential field at the plasma edge (due to Magrid), then invalidates the ambipolar diffusion assumption that the Bohm sheath criteria is based upon.
http://en.wikipedia.org/wiki/Ambipolar_diffusion
No shortcuts.
What I can probably handle is a twotemperature electron distribution. Would that be an acceptable place to start?
Maybe.
Even so though, it looks to me like the Bohm integration doesn't hold because the boundary conditions of the plasma do not meet the Bohm sheath criterion bc's. There is a potential field due to Magrid outside the plasma to consider
http://en.wikipedia.org/wiki/Debye_sheath
"At the sheath edge (ξ = 0), we can define the potential to be zero (χ = 0) and assume that the electric field is also zero (χ' = 0)."
If so, the potential field at the plasma edge (due to Magrid), then invalidates the ambipolar diffusion assumption that the Bohm sheath criteria is based upon.
http://en.wikipedia.org/wiki/Ambipolar_diffusion
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