What is a convex magnetic field?

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D Tibbets
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What is a convex magnetic field?

Postby D Tibbets » Sat Nov 05, 2016 1:20 am

It has been repeatedly stated that the convex magnetic fields in the Polywell eliminates concerns about edge instabilities/ macro instabilities. While this may be true or mostly true, the question I am asking here is how does a bundle of wires not have concave fields between them? To rephrase the question- How do you get a smooth convex B field when there ae concavities in the field between individual windings? If you have 100 wires in a magnet bundle, there is some separation between them, even if it is only a small fraction of a mm due to the varnish insulation. The overall wire bundle magnetic field would be rippled on the surface-concave and convex as it bridged between wires. I do not know the answer but I speculate that several considerations may apply. Any corrections or enlightenment would be appreciated.

If the separation between each wire in the bundle is much less than the gyroradius of the moving charged particles of interest. The average shape of the magnetic field would be dominated by the overall convex field generated by the wire bundle in total. Would the gyroradius variation of the particles (electrons primarily since they have the much smaller gyroradii) in a thermalized plasma be more challenging than a mono energetic plasma?

Also, as the wires/ superconducting tape and possibly other materials in the immediate vicinity of the enclosed magnet would have a permitivity much higher than that in a vacuum or rarefied gas. The internal B fields would thus be much stronger. Would this contrast of the B field inside and outside of the can play a role in presenting a smooth contour B field outside of the can?

In a solenoid with a cylindrical magnetic wire winding, how close do the windings have to be to prevent the concave- convex wobbling in the B field relative to the plasma? I presume a solenoid/ cylinder with parallel walls could be vulnerable to macro instabilities due to local variations in the plasma pressure (due to various plasma characteristics). As such, a small local bulge could occur and then grow due to the favorable energy balance. How much convexity is required to prevent most of these local excursions from occurring? I think cylindrical B fields of this nature are vulnerable to macro instabilities. Would this slightly- to mildly diverging near cylinder help? Combined with separate end opposing ring magnets it starts to look like thee three ring magnet magrid that I have describe before . And, except for the central curvature, like old and discarded cylindrical with end cap designs.If the 'cylinder, or big ring if you prefer, might still allow for central focus (it does as demonstrated by some magnetic modeling I have done before and presented here) provided the length of this curving cylinder is not to long and appropriate B field strengths are applied to this central curving 'cylinder' or fat or oval minor radius magnet and the opposing end magnets. I like a central donut or torus magnet that has a length of ~ 1.5 to 2 times the length/ radius of the opposing end magnets. Also having a greater radius from the midline of the 'cylinder' for the central magnet helps with central focus into the previously mentioned torus, near sphere or dumbell core. I don't know if this central magnet to end magnet length along the cylindrical axis could be pushed past this ratio.

Note that the Lockheed design has elements of this and I originally thought that it was essentially the same. But emphasis on circulating FRC type flows/ recirculation between cusps and lack of a potential well are a major divergence from my concept.

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

prestonbarrows
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Re: What is a convex magnetic field?

Postby prestonbarrows » Thu Dec 29, 2016 5:20 am

The notion of 'convex magnetic fields being stable' arises more clearly from MHD and fluid dynamics than single particle motions. At the temperatures, densities, and field strengths relevant to a reactor, we can think of the gyroradii as negligibly small and the plasma being a fluid that is 'locked' to the magnetic field lines.

A 'flux tube' which follows the magnetic field line and has a cross-sectional area defined by a constant magnetic flux will contain some volume of plasma with some finite total mass and stored thermal energy. The plasma within a given flux tube volume can be approximated by an adiabatic gas. The volume of the flux tube in the infinitesimal limit is given by /delta V in the attached image where /delta /Phi is the constant flux enclosed by the tube and dl is along the magnetic field line.

Consider some perturbation acts to shift the flux tube, while an adjacent 'vacuum flux tube' containing no plasma takes its original place. If the plasma flux tube moves into a location with a lower magnetic field strength, its cross-section (and volume) will expand to maintain the constant flux condition. As the volume expands, the plasma within the flux tube cools adiabatically and gives up thermal energy in the process. This energy in then available to further drive the instability and the flux tube continues to expand, exchanging its thermal energy for kinetic energy. This is conceptually similar to the thermal energy of hot gas inside a piston doing work and being converted to kinetic energy of the piston as it expands and cools adiabatically.

If the magnetic field strength increases in the direction the flux tube is perturbed, the opposite happens. The plasma is compressed causing its temperature and internally stored thermal energy to increase. That energy must come from somewhere and large scale kinetic motion of the perturbation becomes damped.

The same concept happens between two flux tubes containing plasmas at different pressures. i.e. anywhere there is a pressure gradient.

In a reactor, there is always a pressure gradient from the center to the walls. This tends to provide the initial perturbation as pressure gradients create a force which push outwards to lower pressure regions. So, if in any direction the confining magnetic field strength decreases while the pressure decreases also this instability will be possible.


So in your example, even if there are local 'convex' and 'concave' ripples, their effect on the flux tube volume tends to cancel around the closed line integral along dl (the magnetic field line). The magnetic field magnitude and the length of the closed magnetic loop are more important. Again what matters is how the flux tube volume /delta V changes in response to any perturbation, especially to external forces like pressure gradients /grad P, not strictly the local field curvature.

The magnetic field strength always decreases as you move away from a magnet. If the plasma density is higher 'inside' the coil, the instability will be possible (like a tokamak). This is the 'concave' case where if your flux tube is perturbed outward, both the length of the loop increases and the magnitude of B decreases making the integrated volume of the flux tube much larger.

If the plasma density is higher 'outside' the coil, the instability will not be possible (like a polywell etc.) This is the 'convex' case where if your flux tube is perturbed outward, the magnitude of B increases faster than the length of the loop making the integrated volume of the flux tube shrink and resist the perturbation. However, such convex geometries always include some type of cusp field due to Gauss's law; there ain't no such thing as a free lunch.

This is all basically a subset of the more well known Rayleigh Taylor instability. In the plasma world, this is known as the 'balooning instability' and is only strictly true in thermalized plasmas with closed field lines.
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D Tibbets
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Re: What is a convex magnetic field?

Postby D Tibbets » Wed Jan 04, 2017 1:28 am

Thanks for the information.

The relatively tiny B field variations are drowned as the in the gross crowd like behavior as described by fluid dynamics strongly dominates.

As such, the fluid strainer (like a flour sifter) analogy is on safer ground. With the wire mesh strainer/ sifter inverted, thesmall amount of water trapped in the mesh due to surface tension cannot fall due to gravity (substitute for plasma pressure against a B field). Water drops will not fall through the sieve. If the strainer is curved concave to the pressure (which is downward due to gravity) the water will collect into droplets and fall through the mesh, mimicking the edge instabilities/ macro instabilities of a plasma in a confining B field. It is also governed (I think) by potential energy considerations which impede the progression of the water/ plasma droplet. But, with the sieve convex, there are the equivelent of cusps and the water will flow to the edge and fall. There is no free lunch. I suppose having an almost flat sieve with a slight convexity relative to the pressure (downward directed gravity) would slow this flow towards the cusp/ edge. This loosely (very loosely) mimics the high Beta condition- Wiffleball effect. The surface area becomes much greater relative to the edge dimensions. The sieve is larger and almost flat till close to the edge.

With plasma and magnetic fields the mirroring effects and the high Beta adiabatic effects are not represented, but for macro instaiblities considerations it may be a simple and illistrative tool.

I also wonder how close to a flat surface the B field can be to the plasma before microscopic deviations leads to establishment of a concave bubble- which is then energetically favorable for further growth.. Plasma turbulence, chaos theory and interventions may determine the issue. Instead of a cylindrical solenoid, have one that is mildly curved into a convex shape towards the central axis. With opposing end ring magnets it would be similar in part to the Lockheed design and what I call the three ring design described here several years ago. So long as the central ring or curved solenoid magnet is not to long a central focus may still be possible. That is a central spot as opposed to a central axis line of confluent plasma. This differentiates this from the cylindrical designs with end plugging I have seen. It may also have baseline confinement good enough that the convex B field surfaces can be pushed out with adiquate input to establish a high Beta condition with resultant excellent confinement times (within the triple product relationship considerations).

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

hanelyp
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Re: What is a convex magnetic field?

Postby hanelyp » Wed Jan 04, 2017 4:37 pm

For a simpler qualitative model, think of what happens to magnetic field strength when plasma pressure pushes on the magnetic field. In a configuration like the polywell (convex magnetic field) the magnetic field is squeezed between the plasma and coil, getting stronger at the plasma interface as it's pressed. While in a tokomak (concave over much of the field) plasma pressing outward finds a weaker magnetic field.
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D Tibbets
Posts: 2769
Joined: Thu Jun 26, 2008 6:52 am

Re: What is a convex magnetic field?

Postby D Tibbets » Thu Jan 05, 2017 12:30 am

Which begs the question (at least for me) of what are some examples of concave (and closed) B fields confining plasma that has increasing strength as the plasma moves outward?

Is the stellarator a dynamic effort to effectively do this?


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


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