The Universe Is Connected By Birkeland Currents

[Michael Mozina]

If you look at equation 30, and figure 2, the Axial Magnetic Field component Bz, and the Azimuthal Magnetic Field component Bθ follow a sine wave pattern at various radial distances which is what acts to create the concentric rings as various distances. The concentric tube process is certainly a "physical' process with an identified "physical" cause.

It's not a sine wave, but a cylindrical Bessel function.

Well, the Bessel function is cylindrical, but the azimuthal and axial magnetic field components as plotted against the radius in figure 3 of Scott's paper produces two different sine waves that peak at different radii. I suspect it's that feature, along with the different temperature and ionization states of different elements which produces the various concentric tubes, not boundary conditions related to the Bessel function.

Regardless, I'd like you to:
1) show that such a pattern can be obtained from Maxwell's equations with finite boundary conditions (e.g. requiring B = 0 at r > b, where b is some arbitrary distance);
2) if 1 is not possible, at least show that the total current/field over infinite cross section (r goes from 0 to infinity, as is required by the paper) is finite.
I claim that both are impossible. And until someone proves otherwise, Scott's model has nothing to do with reality in my books.

I think that you're oversimplifying or overlooking the actual conditions in plasma and a relevant quote from Alfven is probably in order. From Cosmic Plasma, at the bottom of page 25 and the top of page 26:

In the case of a partially ionized gas mixture, a temperature gradient will cause the radial transport to depend on the ionization potential, so that in general, the elements with the lowest ionization potential are brought closest to the axis . We may expect the elements to form concentric hollow cylinders whose radii increase with ionization potential. Quite generally, it seems likely that for a rather wide range of parameters, a current through a partially ionized plasma is able to produce element separation (IV .3).

Alfven includes a diagram presumably drawn by Markland (to descibe Markland convection) on page 26, figure II.15.

It's not necessarily a boundary condition related to the Bessel function that results in concentric tubes, rather it's related to thermal and ionization potential differences between different elements present in the plasma. I think that you're barking up the wrong tree with respect to the cause of concentric tubes.

[Higgsy]

The problem is that current density (see (49) in the linked paper) is proportional to J_0. Which, since the area rises as r^2, gives you a divergence of the total current at r = infinity.

Understood.

[Higgsy]

If you look at equation 30, and figure 2, the Axial Magnetic Field component Bz, and the Azimuthal Magnetic Field component Bθ follow a sine wave pattern at various radial distances which is what acts to create the concentric rings as various distances. The concentric tube process is certainly a "physical' process with an identified "physical" cause.

It's not a sine wave, but a cylindrical Bessel function.

Well, the Bessel function is cylindrical, but the azimuthal and axial magnetic field components as plotted against the radius in figure 3 of Scott's paper produces two different sine waves that peak at different radii. I suspect it's that feature, along with the different temperature and ionization states of different elements which produces the various concentric tubes, not boundary conditions related to the Bessel function.

Good Heavens!

[paladin17]

I suspect it's that feature, along with the different temperature and ionization states of different elements which produces the various concentric tubes, not boundary conditions related to the Bessel function.
...
I think that you're oversimplifying or overlooking the actual conditions in plasma and a relevant quote from Alfven is probably in order. From Cosmic Plasma, at the bottom of page 25 and the top of page 26:
...
Alfven includes a diagram presumably drawn by Markland (to descibe Markland convection) on page 26, figure II.15.

It's not necessarily a boundary condition related to the Bessel function that results in concentric tubes, rather it's related to thermal and ionization potential differences between different elements present in the plasma. I think that you're barking up the wrong tree with respect to the cause of concentric tubes.

You misunderstood my argument, it seems. And also mixed some apples and oranges in the process. Let's clear some things up.

1) Scott's model is a purely mathematical model (it's a type of solution of Maxwell's equations), and it should be evaluated as such - which I did. In particular, I am concerned with the fact that a) it is only possible in infinitely sized current; b) even in that setup it leads to non-physical results (diverging value of a total current).

2) When I say "boundary conditions", I mean that in a mathematical sense; Maxwell's equations are partial differential equations, so to solve the Cauchy problem you need to set boundary conditions - otherwise the number of solutions is, generally speaking, infinite. Scott assumes his current to be infinite in size (so the boundary where the current goes to zero is at infinity) - which is, by the way, not a problem in mathematical sense (though causes understandable questions in physical sense); but even that leads to infinities as a result (see above: the total current diverges).
So, technically speaking, as a description of infinitely large and infinitely strong current (with an undefined direction at infinity) Scott's model is fairly good. Otherwise, not so much.

3) Since you're quoting Alfven, take a look at these words:

... infinite plasma models, or models with static boundary conditions, are often applied to problems with variable boundary conditions. This gives completely erroneous results.

... magnetic field description is often used in a careless way leading to the neglect of boundary conditions. Infinite models are applied to plasmas with finite dimension, resulting in erroneous conclusions.

In other words, the boundary conditions are essential.

4) Also, take a look at Alfven's solution to the same problem (cylindrical force-free current): see "Cosmical Electrodynamics" (ed. 1, p. 68-70, especially Fig. 3.5) or "Cosmic Plasma" (p. 93-98). Peratt also gives the same solution in his "Physics of the Plasma Universe" (ed. 2, p. 167-169). Please, show me the layers of reversing current there. Hint: you won't find any, because it is not physical, and these guys know what they're doing.

5) Marklund convection arises due to plasma drifts in a current with a temperature gradient. In derivation of his model, Scott never even considers temperature or conductivity etc. So this process is unrelated to the problem we're discussing.
In fact, Scott's reversing current layers might actually shut the convection down, since his model contains areas with reversing directions of E and B vectors, which (if a starting condition is a homogeneous mixture of ionized gases, plus a temperature gradient) could cause a certain ion species to drift inwards in one area and outwards in an adjacent one. In Marklund's consideration (just as in Alfven's, Peratt's etc.) there are no reversing current layers. They only exist in Scott's model, which (for the two reasons I've stated in other post above) has nothing to do with reality.

[Michael Mozina]

I suspect it's that feature, along with the different temperature and ionization states of different elements which produces the various concentric tubes, not boundary conditions related to the Bessel function.
...
I think that you're oversimplifying or overlooking the actual conditions in plasma and a relevant quote from Alfven is probably in order. From Cosmic Plasma, at the bottom of page 25 and the top of page 26:
...
Alfven includes a diagram presumably drawn by Markland (to descibe Markland convection) on page 26, figure II.15.

It's not necessarily a boundary condition related to the Bessel function that results in concentric tubes, rather it's related to thermal and ionization potential differences between different elements present in the plasma. I think that you're barking up the wrong tree with respect to the cause of concentric tubes.

You misunderstood my argument, it seems.

I assumed that was the case to begin with when I pointed out that I'm pretty sure that I'm not following your argument properly. I'm trying. Please bear with me a bit.

And also mixed some apples and oranges in the process. Let's clear some things up.

1) Scott's model is a purely mathematical model (it's a type of solution of Maxwell's equations), and it should be evaluated as such - which I did.

Hmm. Well, ok, sure it's ultimately a mathematical model, but it's physically related specifically to plasma and the movements of plasma in current carrying environments. I'm not sure we can ignore the physics aspects entirely and properly evaluate it.

In particular, I am concerned with the fact that a) it is only possible in infinitely sized current; b) even in that setup it leads to non-physical results (diverging value of a total current).

It might help me to understand why you believe it only applies to infinitely sized currents by citing specific formulas that imply this. I don't follow that argument.

2) When I say "boundary conditions", I mean that in a mathematical sense; Maxwell's equations are partial differential equations, so to solve the Cauchy problem you need to set boundary conditions - otherwise the number of solutions is, generally speaking, infinite. Scott assumes his current to be infinite in size (so the boundary where the current goes to zero is at infinity) - which is, by the way, not a problem in mathematical sense (though causes understandable questions in physical sense); but even that leads to infinities as a result (see above: the total current diverges).

I'm still unclear why you assume that his current must be infinite in size in all cases. Some of the equations might assume the potential integrates from zero to infinity, but that's not necessarily implying that the current is necessarily infinite. Could you cite the specific formula that makes that infinite current a *necessary* requirement? It's possible I'm missing something important but you'll have to be a bit more specific for me to better understand your argument.

So, technically speaking, as a description of infinitely large and infinitely strong current (with an undefined direction at infinity) Scott's model is fairly good. Otherwise, not so much.

Well, again, I'll need you to cite specific formulas which your argument is based on. Integration to infinity doesn't necessarily *require* current to be infinite.

3) Since you're quoting Alfven, take a look at these words:

... infinite plasma models, or models with static boundary conditions, are often applied to problems with variable boundary conditions. This gives completely erroneous results.

... magnetic field description is often used in a careless way leading to the neglect of boundary conditions. Infinite models are applied to plasmas with finite dimension, resulting in erroneous conclusions.

In other words, the boundary conditions are essential.

4) Also, take a look at Alfven's solution to the same problem (cylindrical force-free current): see "Cosmical Electrodynamics" (ed. 1, p. 68-70, especially Fig. 3.5) or "Cosmic Plasma" (p. 93-98). Peratt also gives the same solution in his "Physics of the Plasma Universe" (ed. 2, p. 167-169). Please, show me the layers of reversing current there. Hint: you won't find any, because it is not physical, and these guys know what they're doing.

5) Marklund convection arises due to plasma drifts in a current with a temperature gradient. In derivation of his model, Scott never even considers temperature or conductivity etc. So this process is unrelated to the problem we're discussing.
In fact, Scott's reversing current layers might actually shut the convection down, since his model contains areas with reversing directions of E and B vectors, which (if a starting condition is a homogeneous mixture of ionized gases, plus a temperature gradient) could cause a certain ion species to drift inwards in one area and outwards in an adjacent one. In Marklund's consideration (just as in Alfven's, Peratt's etc.) there are no reversing current layers. They only exist in Scott's model, which (for the two reasons I've stated in other post above) has nothing to do with reality.

I'll spend some time looking up the material you suggested to help me better understand your argument. In the meantime if you could be a bit more specific about which exact formulas you're concerned about, that would definitely help me to better follow your argument.

[Michael Mozina]

By the way Paladin, have your shared your concerns with Dr. Scott yet? If so, I'm curious to know how he responded.

[JP Michael]

I recall reading mention of the importance of elemental ionisation potential in one of Barry Setterfield's papers, section I.H and following [1]:

Where currents flow in ionized or partially ionized plasma filaments, a separation of elements may occur. With a variety of ions in a filament, there tends to be a preferential, radial transportation of ions. The elements with lowest ionization potential are brought closest to the current axis. Peratt points out that the most abundant elements found in cosmic plasma will be sorted into a layered structure in plasma filaments. He states: "Helium will make up the most widely distributed outer layer; hydrogen, oxygen and nitrogen should form the middle layers; and iron silicon and magnesium will make up the inner layers. Interlap between the layers can be expected and, for the case of galaxies, the metal-to-hydrogen ratio should be maximum near center and decrease outwardly" (citing Peratt [2])

The order of elements going from the center of the filament outwards using the approximate first ionization potential in electron volts (eV) is then as follows: the radioactive elements Rubidium, Potassium, Radium, Uranium, Plutonium and Thorium (4.5 to 6 eV); Nickel, Iron and metals (7.7 eV); Silicon (8.2 eV); Sulfur and Carbon (10.5 to 11.3 eV); Hydrogen and Oxygen (13.6 eV); Nitrogen (14.5 eV); Helium (24.5 eV). As Peratt points out, this mechanism provides an efficient means to accumulate matter within a plasma (citing Peratt, [3]). This property of plasma filaments, which causes different chemical elements to distribute themselves radially according to their ionization potentials, was initially studied in detail by Marklund, and is now called Marklund convection (citing Marklund, [4]).

Marklund states: "In my paper in Nature the plasma convects radially inwards, with the normal [predicted] velocity, towards the center of a cylindrical flux tube. During this convection inwards, the different chemical constituents of the plasma, each having its specific ionization potential, enter into a progressively cooler region. The plasma constituents will recombine and become neutral, and thus no longer be under the influence of the electromagnetic forcing. The ionization potentials will thus determine where the different species will be deposited, or stopped in their motion." Ionization potential charts show where, in a layered filament, the highest concentration of any element will probably be found. The velocity, v, with which the various ions will drift towards the center is given by (citing Peratt, [2])

v = (E x B)/ B^2 (2)

Here, E is the electric field strength, B is the magnetic flux density or magnetic induction, and the electromagnetic force on ions causing them to drift is given by (E x B). It is generally true that the ion drift velocity in a system is given by the ratio of the electric field to the magnetic field, that is:

v = E/B (3)

For electromagnetic waves, v = c, where c is the velocity of light. In plasma out in space, the typical drift velocity of ions can often approach the speed of light, or at least a significant fraction of it. The formula for the rate of accumulation of ions in a filament is given by dM/dt which is defined as (citing Peratt, [5]):

dM/dt = (2πr)^2 ρ[E/(μI) (4)

where r is the radius of the filament, E is the electric field strength, I is the electric current, μ is the magnetic permeability of the vacuum and ρ is the number density of ions.

The next section, I.I, Setterfield discusses interacting currents:

Another important property is that Birkeland currents interact with each other. Two parallel Birkeland currents moving in the same direction will attract each other with a force that is inversely proportional to the distance between them (citing Martin & Connor [6]). This contrasts with the gravitational force which is inversely proportional to the square of the distance. In fact, in plasma, electromagnetic forces can exceed gravitational forces by a factor of 1039. Furthermore, even in neutral hydrogen regions of space where the ionization is as low as 1 part in 10,000, electromagnetism is nevertheless still about 107 times stronger than gravity (citing Martin & Connor [6]). Not only is this electromagnetic attraction much stronger than gravity, the attractive force is also proportional to the strength of each current multiplied together (citing Martin & Connor [6]). Thus stronger currents result in even stronger attractive forces. If the currents are moving in opposite directions, the same proportionalities hold but the force is repulsive. The same holds true for electric currents in parallel wires as Ampere first demonstrated in 1820. This can be expressed mathematically. If the attractive or repulsive force is δF, on a length δl of either current whose distance apart is r, with the currents being I[sub]1 and I[sub]2 respectively, then we can write (citing Langmuir, Found & Dittmer [7]):

δF / δl = (I[sub]1 I[sub]2 / r)(μ/2π) (5)

In equation (5) the quantity μ is again the magnetic permeability of the vacuum. For a comment on this, please refer to the discussion following equation (14) and the preamble to equation (51).

[[Note my addition of 'sub' in eq. (5) due to the forum having removed super/subscript features]]

[1] B. Setterfield, "Reviewing A Plasma Universe with the Zero Point Energy," Journal of Vectorial Relativity 3(3):1-29, Sept. 2008.
[2] A.L. Peratt, IEEE Transactions On Plasma Science, Vol. PS-14:6 (December 1986), pp. 763-778.
[3] A.L. Peratt, Physics of the Plasma Universe, Chapter 4 Section 4.6.3, Springer-Verlag (1992).
[4] G.T. Marklund, Nature, 277 (1st Feb. 1979), pp. 370-371.
[5] A.L. Peratt, Astrophysics & Space Science, 256 (1998), pp. 51-75.
[6] S.L. Martin & A.K. Connor, Basic Physics, 8th Edition, Vol. 2, pp. 721-724, Whitcombe & Tombs Pty.Ltd.,
Melbourne (1959).
[7] I. Langmuir, C.G. Found & A.F. Dittmer, Science 60 (1924), p.392.

[paladin17]

It might help me to understand why you believe it only applies to infinitely sized currents by citing specific formulas that imply this. I don't follow that argument.

He explicitly states that

No boundary condition at any non-zero value of r is introduced.

I.e. r goes from 0 to infinity. In the very next sentence he mentions that

There will be, in all real currents in space, a natural limit, r = R, to the extent of the current density j(r).

However, he does not discuss how the solution would change in this case. Which returns us to my point 1) in this post.
People who used finite boundary conditions (Alfven/Faelthammar, Peratt etc. - see citations above) did not get any current reversals. Hence the claim that I'm making in my posts here.

By the way Paladin, have your shared your concerns with Dr. Scott yet? If so, I'm curious to know how he responded.

No, I didn't. Nor do I plan to.

[Michael Mozina]

It might help me to understand why you believe it only applies to infinitely sized currents by citing specific formulas that imply this. I don't follow that argument.

He explicitly states that

No boundary condition at any non-zero value of r is introduced.

I.e. r goes from 0 to infinity. In the very next sentence he mentions that

There will be, in all real currents in space, a natural limit, r = R, to the extent of the current density j(r).

However, he does not discuss how the solution would change in this case. Which returns us to my point 1) in this post.
People who used finite boundary conditions (Alfven/Faelthammar, Peratt etc. - see citations above) did not get any current reversals. Hence the claim that I'm making in my posts here.

By the way Paladin, have your shared your concerns with Dr. Scott yet? If so, I'm curious to know how he responded.

No, I didn't. Nor do I plan to.

I'm still reading through the materials that you suggested, so I may have to more to say once I'm done. In the meantime however let me make a couple of points.

By Scott *not* imposing boundary conditions on r, and integrating from 0 to infinity, he's not requiring or describing infinite current, just infinite space/distance. If one BK model imposes a boundary condition on r, then the movement pattern observed over that limited distance r might only find one plasma tube moving in a single direction, whereas another model with a larger range of r values might show current movement patterns in different directions at different r values. That's essentially the case here as far as I can tell.

I would also encourage you to email Dr. Scott about your concerns, since I'm sure that he understands the nuances of his own model *far* better than I do. By the way, I've emailed Dr. Scott with questions in the past and he's always been most gracious about taking the time to answer them, even my basic questions. He doesn't bite.

As far as I can tell thus far, your criticism isn't valid because you're comparing apples to oranges. You're comparing models which *do* impose r value boundary conditions to another model which does not impose such restrictions. It's therefor possible (and logical) that both models might produce the same movement pattern for plasma at the bounded r value of the bounded model, and yet the unbounded model might produce more information related to different concentric tubes moving in different patterns at higher r values. You're also implying that we can ignore the physical processes and features of plasma (like Murklund convection) when it's highly likely that it's the specific physical features of plasma that are responsible for the various concentric tubes in the first place. In short, I don't think your criticisms seem to be valid. I don't see how its even valid to require Scott to impose boundary conditions on r, when his entire concept is based on what happens in plasma at *various* r values. If he imposed a boundary condition on r, then he might come up with just one tube moving in one direction. By not imposing boundary conditions he might come up with concentric tubes moving in different directions. That seems to be exactly the case here.

[paladin17]

By Scott *not* imposing boundary conditions on r, and integrating from 0 to infinity, he's not requiring or describing infinite current, just infinite space/distance. If one BK model imposes a boundary condition on r, then the movement pattern observed over that limited distance r might only find one plasma tube moving in a single direction, whereas another model with a larger range of r values might show current movement patterns in different directions at different r values. That's essentially the case here as far as I can tell.

The fact that the total current in this model is infinite (actually "oscillates" from -infinity to +infinity) is guaranteed by the expression (49) in the paper. Current density depends on J_0, so total current (density times area) over infinite cross section is infinite.
I disagree about the models. They are scalable (see parameter alpha in the expressions), so it doesn't matter what size you choose. If you impose a boundary condition (to get a physical result) you don't get current reversals. If you don't impose it (to get Scott's result), you get them. End of story (as far as I can tell).

You're comparing models which *do* impose r value boundary conditions to another model which does not impose such restrictions.

Exactly. The difference is that the former yield physically meaningful result, while the latter doesn't.

It's therefor possible (and logical) that both models might produce the same movement pattern for plasma at the bounded r value of the bounded model, and yet the unbounded model might produce more information related to different concentric tubes moving in different patterns at higher r values.

No, that's not how partial differential equations work. The solution depends strongly on boundary conditions. See also quotes from Alfven that I gave previously.

You're also implying that we can ignore the physical processes and features of plasma (like Murklund convection) when it's highly likely that it's the specific physical features of plasma that are responsible for the various concentric tubes in the first place.

Yes, we can ignore that, since it's an unrelated problem in the first place. Marklund convection doesn't lead to current reversals, so it doesn't support Scott's model, so it can be disregarded with respect to this discussion. Actually, as I've noted previously, the phenomenon itself was derived starting from the assumption of no current reversals. So I fail to see why you keep bringing it up.

[Michael Mozina]

By Scott *not* imposing boundary conditions on r, and integrating from 0 to infinity, he's not requiring or describing infinite current, just infinite space/distance. If one BK model imposes a boundary condition on r, then the movement pattern observed over that limited distance r might only find one plasma tube moving in a single direction, whereas another model with a larger range of r values might show current movement patterns in different directions at different r values. That's essentially the case here as far as I can tell.

The fact that the total current in this model is infinite (actually "oscillates" from -infinity to +infinity) is guaranteed by the expression (49) in the paper. Current density depends on J_0, so total current (density times area) over infinite cross section is infinite.

Integration over infinite distance of a *finite* current/number of particles doesn't lead to infinite current. It ultimately depend on the number of particles present, and the amount of current present.

Yes, we can ignore that, since it's an unrelated problem in the first place. Marklund convection doesn't lead to current reversals, so it doesn't support Scott's model, so it can be disregarded with respect to this discussion. Actually, as I've noted previously, the phenomenon itself was derived starting from the assumption of no current reversals. So I fail to see why you keep bringing it up.

Though I haven't finished all the materials you've suggested yet, my position hasn't changed in this regard. The location of ions does seem to be influenced by their ionization state, and even Aflven suggested that concentric tubes might form this way. It's also entirely possible that the *limit* on the r leads to more limited results, and other results are possible at greater r values, so you're still comparing apples to oranges IMO.

I'd encourage you again to contact Scott with your concerns. I would hate to stick any words in his mouth over this issue, but I've heard him specifically discuss Marklund convection and discuss the changing EM fields at various r values, so I think I'm on pretty good footing. I don't think it's reasonable to ignore the physical processes in plasma that help drive particle/elemental separation, and I don't think it's fair to assume that a limited r value study is necessarily going to reveal *all* possible r value possibilities in terms of particle movement. I think you're making too many assumptions about Scott's work.

[paladin17]

Integration over infinite distance of a *finite* current/number of particles doesn't lead to infinite current. It ultimately depend on the number of particles present, and the amount of current present.

This is absolutely correct. And also has nothing to do with Scott's model.
He does not demonstrate that the current in his model is finite, and I claim that it is impossible to demonstrate.

Though I haven't finished all the materials you've suggested yet, my position hasn't changed in this regard. The location of ions does seem to be influenced by their ionization state, and even Aflven suggested that concentric tubes might form this way. It's also entirely possible that the *limit* on the r leads to more limited results, and other results are possible at greater r values, so you're still comparing apples to oranges IMO.

Quite the contrary, you keep bringing irrelevant problems up. Concentric tubes of different elements is not the same as concentric shells of counter-directed current. Scott doesn't even talk about particles in the first place. He just considers currents and fields, in a continuous (non-quantized) perspective.

[Michael Mozina]

Integration over infinite distance of a *finite* current/number of particles doesn't lead to infinite current. It ultimately depend on the number of particles present, and the amount of current present.

This is absolutely correct. And also has nothing to do with Scott's model.
He does not demonstrate that the current in his model is finite, and I claim that it is impossible to demonstrate.

Though I haven't finished all the materials you've suggested yet, my position hasn't changed in this regard. The location of ions does seem to be influenced by their ionization state, and even Aflven suggested that concentric tubes might form this way. It's also entirely possible that the *limit* on the r leads to more limited results, and other results are possible at greater r values, so you're still comparing apples to oranges IMO.

Quite the contrary, you keep bringing irrelevant problems up. Concentric tubes of different elements is not the same as concentric shells of counter-directed current. Scott doesn't even talk about particles in the first place. He just considers currents and fields, in a continuous (non-quantized) perspective.

I think at this point we're just going to have to agree to disagree. I think you're missing the point with respect to r values. Scott's point is to demonstrate that the magnetic field patterns and particle movement patterns can vary at different r values, not to suggest that all currents must necessarily be infinite. Sure, if r really is extended into infinity in the *real* world, that may indeed result in infinite current but that isn't the point of the model. It's simply being used to demonstrate that various r values result in different magnetic field patterns, and different particle movement patterns within the same Birkeland current.

Furthermore, I don't believe that ionization aspects of plasma and Marklund current features are irrelevant, or that they can simply be ignored with respect to generating tubes of different concentric sizes. Different elements will ionize differently in the presence of current and even Alfven suggests that this might be responsible for the formation of current carrying tubes of different sizes and generate elemental separation within the same Birkeland current.

[Higgsy]

Integration over infinite distance of a *finite* current/number of particles doesn't lead to infinite current. It ultimately depend on the number of particles present, and the amount of current present.

This is absolutely correct. And also has nothing to do with Scott's model.
He does not demonstrate that the current in his model is finite, and I claim that it is impossible to demonstrate.

Though I haven't finished all the materials you've suggested yet, my position hasn't changed in this regard. The location of ions does seem to be influenced by their ionization state, and even Aflven suggested that concentric tubes might form this way. It's also entirely possible that the *limit* on the r leads to more limited results, and other results are possible at greater r values, so you're still comparing apples to oranges IMO.

Quite the contrary, you keep bringing irrelevant problems up. Concentric tubes of different elements is not the same as concentric shells of counter-directed current. Scott doesn't even talk about particles in the first place. He just considers currents and fields, in a continuous (non-quantized) perspective.

I think at this point we're just going to have to agree to disagree. I think you're missing the point with respect to r values. Scott's point is to demonstrate that the magnetic field patterns and particle movement patterns can vary at different r values, not to suggest that all currents must necessarily be infinite. Sure, if r really is extended into infinity in the *real* world, that may indeed result in infinite current but that isn't the point of the model. It's simply being used to demonstrate that various r values result in different magnetic field patterns, and different particle movement patterns within the same Birkeland current.

Furthermore, I don't believe that ionization aspects of plasma and Marklund current features are irrelevant, or that they can simply be ignored with respect to generating tubes of different concentric sizes. Different elements will ionize differently in the presence of current and even Alfven suggests that this might be responsible for the formation of current carrying tubes of different sizes and generate elemental separation within the same Birkeland current.

I've been following this thread with some interest, watching Paladin's patient explanations and Michael's refusal to follow them. It's interesting, because it's completely unlike any conversation that I might have with colleagues about the same paper - who all, right at the top of the thread, would have said "Yeah, I see that, the solution is not unique and is unphysical. Time to move on."

I have to lay most of the blame for the lack of progress at Michael's door, because there are some fairly fundamental things that he doesn't seem to be aware of, or is choosing not to acknowledge. Much of what I write below is a bare repeat of Paladin's points.

1) The plain fact that the theoretical portion of the linked paper treats only solutions to the Maxwell equations under certain conditions (field aligned current, no variation in B axially or azimuthally, but only radially etc), and refers only to continuous magnetic fields and currents. It does not discuss particles at all in arriving at the solution, so any consideration of plasma, elemental separation or ionisation state separation is irrelevant to considering the theoretical results of this paper.

2) The need to set boundary conditions when solving partial differential equations which describe a physical problem in order to arrive at a unique and physical solution. This issue, known as the Cauchy problem, is taught as part of undergraduate courses on mathematical methods for physicists, so I can see why Michael might not be familiar with it. It arises from the fact that the general solution for partial differential equations contains arbitrary functions and, in physical problems, the arbitrary functions must be chosen by setting appropriate boundary conditions. In this case an appropriate boundary condition might be for B=0 on some finite r and for ∂B_θ/∂r=∂B_z/∂r=0 on the same surface.

Michael has struggled to understand the meaning and importance of boundary conditions in this context, and is yet to accept the need for them. He has repeatedly referred to boundary conditions as being somehow a limitation of the solution rather than a necessary condition for finding a unique solution.

3) The solution proposed in the paper is unphysical in that variables (specifically the current integrated over the area) which have physical meanings do not converge as r goes to infinity.

4) Less important for the consideration as to whether the solution in the paper actually represents a physical solution, but quite important for Michael's understanding, is his repeated claim, even after being corrected, that the solution produces two sine waves. That calls into question Michael's understanding of the paper and his knowledge of the properties of Bessel functions. I also wonder about how closely he is following the mathematical argument, as a trigonometric function does not appear anywhere (except in the asymptotic expansion for large r).

Given these points, it is a wonder how anyone can continue to argue the case for the linked paper, as a physical solution to the Maxwell equations. And yet the discussion has gone on for more than a week with no progress. I can't flatter myself that this will in any way break the log jam.

[Michael Mozina]

I think this is a extremely timely new video on this topic which delves into all the various questions/concerns (and a few more) that we've discussed in this thread. It talks about Scott's choice to not include boundary conditions, and the reasons for and limitations of that approach. It also discusses Marklund convection, and the problems that it presents in Scott's model, as well as points out the fact that Scott's model focuses exclusively upon ion movement without really explicitly dealing with, and describing electron movement patterns and influences.

https://youtu.be/mburLnMO4_I

Suffice to say that there are indeed areas of Scott's Bessel function approach that could use additional work and deserve additional work. It does note that there's a valid and legitimate scientific reason for not necessarily trying to apply specific boundary conditions at various r values because various shells have different Azimuthal and Axial Magnetic field influences on different shells at different radii, but it also rightfully notes that realistically speaking, the filaments of spacetime do indeed have visible boundaries. It discusses the laboratory support of concentric ion movement patterns seen in Peratt's work, and it covers this topic wonderfully IMO.

I think the best way to sum up the Bessel function model of Birkeland currents, is that while it's a great step in the right direction, and it enjoys laboratory support in terms of the movement patterns observed in the lab, there are areas of Scott's model that probably could use additional work. Is that really a "bad' thing however, or just a natural sort of scientific progression that we would expect to see at this point in time? I would argue that it's the later.

It should be noted that had Scott arbitrarily chose a specific r value, it may not have described the full range of different movement patterns of different shells associated with various Azimuthal and Axial Magnetic field changes that affect each shell differently. One could logically make the argument that once the model becomes repetitive in terms of various shell movement patterns and magnetic field influences, that would probably be a good place to apply a boundary condition. As I've been trying to suggest however, picking an *arbitrary* location to impose a boundary condition might unnecessarily and unwisely oversimplify the model to the point of it not fully describing each possible shell configuration and movement pattern associated with that shell.

That was a great (and fair) video IMO. It not only discusses the strengths of Scott's model, but also it's limitations, and it includes visual diagrams that make it much easier to understand and visualize. It's very well done.