G 

The charge sheath vortex develops within a large charge cloud where the repulsion between the charges is cancelled out,
1 ) Two stationary particles carrying the same electrical charge will repel each other, but within an electric field they will form dipoles that will experience an attractive force.
2 ) In a large charge cloud where the repulsion between the charges is the same in all directions, this force will cancel out
3 )  Two streams of dipoles with the same field charge  moving in parallel in a large charge cloud will also develop a force of attraction through their electromagnetic interaction.
4) An electron cascade of electrons briefly following a free path through the magnetic field will briefly travel at very much higher velocities and contribute to a greatly intensified electromagnetic field.
By a quirk of history, diagrams of electrical current show the arrow for current direction for the movement of positive charge. This makes no difference for the charge sheath, but can be a little confusing for students of electricity at first.

The arrow on field lines is shown pointing to the south magnetic pole. The direction shown by a compass.
The north pole on  a magnetic compass points to a south magnetic pole  -(At the geographic north pole!)

Some of the following images come from the hyperphysics site.

This shows the magnetic field produced by a positive charge particle moving. Note the way the thumb placed in the direction of movement and the fingers showing the magnetic field.


If several charged particles move in parallel, the magnetic forces between them draw them together.
We know from Z pinch experiments that the electromagnetic forces that draw the particles togther is greater than the electrostatic charge that pushes them apart.

When a moving charge passes through a magnetic field it experiences a force which is perpendicular to both the magnetic field and the velocity.  - shown by the red dotted line in the following diagram. As the particle responds to the force, by moving upwards, the force continues to act perpendicular to the new path, and so the particle will be forced into a circular path if it remains within the magnetic field..



But the moving charge particle also produces a magnetic field. Note that the direction of the magnetic field produced by the charge particle  (arrows point to the south pole)  is the opposite direction  in relation to the  circular path of the charged particle as in the diagram above. The charge particles are therefore cancelling the magnetic field  which caused the charge particle to follow a circular path.

This would seem to indicate that  a plasma stream should not produce a self confining vortex. But it is not this force that is at work in the charge sheath vortex. The charge particles are not turning in response to moving through an external  magnetic field!

The charge sheath vortex develops within a large charge cloud where the repulsion between the charges is cancelled out,
1 ) Two stationary particles carrying the same electrical charge will repel each other, but within an electric field they will form dipoles that will experience an attractive force.
2 ) In a large charge cloud where the repulsion between the charges is the same in all directions, this force will cancel out
3 )  Two streams of dipoles with the same field charge  moving in parallel in a large charge cloud will also develop a force of attraction through their electromagnetic interaction.
4) This can develop into an electron cascade where fast moving free electrons briefly follow the circular path round the vortex and greatly increase the magnetic field strength of the vortex.

Consider the case where a fast moving stream of positive particles forms a  loop in the absence of an externally applied magnetic field. There is a powerful force of attraction between the particles moving in parallel as in B above.
If a charged particle strays inwards into the inner field, the magnetic force on the charged particle will  subject the particle to a force that pushes it back out into the sheath as discussed in C and D above. If the charged particle strays outwards into a region where the field is in the opposite direction, the force it is subject to will push it back into the sheath.
Note that the charge particles have to be very fast moving in order to sustain this vortex. At lower speeds the charge repulsion will be greater than the electromagnetic attraction, unless the vortex forms in a small region of a larger uniform charge cloud.



The flow is held as a coherent stream by the attraction between forces moving in parallel. Any instability in velocity will cause greater magnetic fields in the faster region than the slower regions.
The faster region with the greater magnetic field will be pinched  more, speeding up the flow, which increases the pinch and so on becoming longer and thinner. But the extension in length cannot be accommodated in a straight line. It can only force the stream to form a loop.

The forces between the charge particles within the charge sheath are balanced by the charge field in the surrounding region in a large cloud, so the force of repulsion between the charges is zero  -only the electromagnetic field remains. If an electron cascade takes place then the vortex will be stable even in a small region of plasma.


Within a non neutral plasma, if the plasma is travelling in a straight line,  the inwards forces on both sides of the plasma are equal, but if a kink starts to form, the plasma on the outer surface is travelling faster than the plasma on the inner surface. The inward force on the outer surface is therefore greater than the outward force on the inner surface.
  The forces due to charge are balanced by the forces in the surrounding charge cloud.
As soon as a loop establishes itself in a flowing plasma, the force of attraction between charge flowing in parallel will wind the plasma onto the end of the coil,  producing a long vortex filament.
The outer surface of the vortex filament continues to rotate faster than the inner surface, so the inward force on the outer surface remains stronger than the outward force on the inner surface. This ensures that the plasma, once formed into a vortex will remain in the vortex.



The plasma sheath is compressed between the powerful inward and outward forces, accelerating and generating yet more powerful forces. Since the outer force pushing inwards always remains the stronger, the vortex will compress and narrow- until the forces generated by the opposing sheath wall spinning in the opposite direction prevent further compression.
The force needed to turn the particles into the curve  will also increase as the velocity increases and radius of curvature increases and will put a limit on the minimum radius the vortex can achieve.
Within the centre of the charged sheath vortex  the fields reinforce each other to produce a powerful solenoidal field.

The charge sheath vortex develops within a large charge cloud where the repulsion between the charges is cancelled out,
1 ) Two stationary particles carrying the same electrical charge will repel each other, but within an electric field they will form dipoles that will experience an attractive force.
2 ) In a large charge cloud where the repulsion between the charges is the same in all directions, this force will cancel out
3 )  Two streams of dipoles with the same field charge  moving in parallel in a large charge cloud will also develop a force of attraction through their electromagnetic interaction.

If there is a significant population of free electrons within the rotating plasma, then these will move in the direction of rotation at much greater velocities than the bulk flow of the plasma, and contribute a very large component to the solenoidal or torroidal electromagnetic fields. This will make the charge sheath vortex stable even in a small volume of charged plasma.

When you have studied the pages you are welcome to join the discussion, but please note that any claims you make for or against these theories should be backed by scientific reasoning that can be developed from first principles.