Conceptually a plasma spheromak is a closed plasma sheet formed by a closed spiral of plasma hose in the plane of the plasma sheet. The direction of the plasma hose axis within the plasma sheet conforms to the toroid surface curvature.
The plasma hose current causes both toroidal and poloidal magnetic field components.
A spheromak is cylindrically symmetric about the spheromak main axis and is mirror symmetric about the spheromak's equatorial plane. The plasma sheet has a net charge Qs that is uniformly distributed over the plasma hose length Lh. This distributed charge causes a spherically radial electric field outside the plasma sheet and a cylindrically radial electric field inside the plasma sheet. The center to center plasma hose spacing Dh is smaller in the spheromak core than at the spheromak periphery. Hence the charge per unit area on the plasma sheet is higher in the spheromak core than at the spheromak periphery.
In the central core of the spheromak the electric field is zero and the purely poloidal magnetic field points along the spheromak's main axis of symmetry. In the region enclosed by the plasma sheet the magnetic field is purely toroidal and the electric field is cylindrically radial. In the region not enclosed by the plasma sheet the magnetic field is purely poloidal and the electric field is spherically radial. The charge circulates in the narrow low magnetic field region at the interface surface between the toroidal and poloidal magnetic fields.
A spheromak can be mathematically modelled as a closed single layer plasma hose spiral in the shape of a toroid. This plasma sheet is formed from a long length Lh of plasma hose. The plasma hose axial direction gradually changes over the surface of the toroid. The behavior of plasma hose is discussed on the web page titled: PLASMA SHEET PROPERTIES.
Photographs of experimental spheromaks within cylindrical metal enclosures show that the experimental spheromak cross section is not round. The following diagram shows the approximate cross sectional shape of an experimentally observed spheromak.
Thus R takes only positive values whereas H takes both positive and negative values.
A spheromak is a cylindrically symmetric closed charge sheet formed from spiraling charge enclosing a toroidal shaped volume. However, for a real spheromak in a vacuum chamber the toroid cross section is not round. The radius of curvature on the outside of the spheromak near the equatorial plane is greater than the radius of a coaxial sphere. Near the equatorial plane the real spheromak core radius is nearly constant. A real spheromak exhibits distinct small radius corners on the charge sheet at the position of the maximum spheromak axial length.
A spheromak has important distinct geometrical dimensions.
An experimental spheromak photographed in a laboratory differs from an ideal spheromak in free space in two important respects.
Consequently the field energy density outside the spheromak on the equatorial plane is cylindrical rather than spherical. This change in external field energy density causes a reduction in the spheromak outside radius Rs and reduces the position stability of the spheromak outside wall.
is not exactly true. This issue causes spheromak shape distortion, especially in the necks of the spheromak where the electric fields from opposite sides of the spheromak are partially cancelling and the consequently reduced energy density is supplemented by the poloidal magnetic field energy density.
Lt = length of each purely toroidal turn.
After an element of charge moving along the plasma hose has passed through the spheromak core hole Nt times and has circled around the main axis of spheromak symmetry Np times, it reaches the point in the plasma hose closed path where it originally started.
There are Nt parallel plasma hoses that go through the equatorial plane in the central core of the spheromak and hence form the spheromak core walls.
This equation establishes a relationship between the number of plasma ions Ni, the number of plasma free electrons Ne and the free electron velocity Ve. This equation is a condition for plasma sheet existence and hence is a condition for plasma spheromak existence.
Note if the plasma hose charge per unit length is uniform then |Ve| is constant and hence the electron kinetic energy is constant along the plasma hose. Hence the plasma sheet is an equipotential surface.
the spheromak magnetic field is almost entirely poloidal and the spheromak is known as a FRC (Field Reversed Configuration).
The toroidal magnetic field can point either Clock Wise (CW) or Counter Clock Wise (CCW) around the spheromak's central poloidal magnetic field. Hence a spheromak has two possible distinct magnetic states of equal energy.
the poloidal magnetic field components cancel to zero.
Thus in numerical evaluation of plasma sheet properties Vi is generally negligibly small.
Thus the effect of a positive radial electric field is to reduce the radius of curvature of ions orbiting the spheromak central axis.
Thus the effect of a positive radial electric field is to increase the radius of curvature of electrons orbiting the spheromak central axis.
and there is no radial electric field.
However, when the electrons and ions are following the same path in opposite directions collisions between electrons and ions cause the electrons to lose a bit energy and the ions to gain a bit energy. This change in particle energy causes the electron and ion paths to separate sufficiently that the electron and ion streams no longer collide. However, an electric field forms that tends to increase the electron path radius of curvature and decrease the ion path radius of curvature, causing maintenance of plasma hose. Over the life of the spheromak, as the free electrons lose energy to the ions the radial electric field E increases to maintain the plasma hose.
The spheromak ceases to exist.
The value of B is this equation is at a magnetic field minimum location where it is impractical to directly measure the magnetic field. However, we may be able to approximately infer the value of B from the spheromak size.
This low value suggests that for a plasma spheromak the magnetic field energy completely dominates the electric field energy. Experimental evidence suggests that the same is true in an atomic particle spheromak.
Hence in a spheromak both electrons and ions have the same momentum magnitude of opposite sign so that the net momentum is zero, In a random plasma the energy rather than the momentum is equally distributed over the particles.
and during spheromak decay the free electrons lose energy to the ions as well as to neutral particles.
However, note that the ion doppler signal may be confused by the thermal component of ion motion, which will generate emission line broadening. Further confusion will occur if there is any uncertainty with respect to the ion mass.
Initially use a spheromak formed from He-4 because, unlike hydrogen that can exist as either molecular or atomic hydrogen ions, there is no uncertainty about the ion mass. Assume a He-4 spheromak with a free electron kinetic energy of 25 eV.
Assume that the atoms are singly ionized (He+ ions).
= 40.05 X 10^-19 J.
This velocity causes a small frequency shift, above or below Fe, but this frequency shift may easily be lost in the thermal broadening.
If ions with this velocity are present they should be easy to detect.
This frequency shift is at least comparable to the thermal line broadening.
Thus the underlying assumption that a spheromak is composed of a plasma sheet can be experimentally demonstrated. At low free electron kinetic energies it may be difficult to separate the ion doppler frequency shift due to spheromak particle motion from the line broadening due to thermal ion motion. However, for compressed spheromaks with 400 eV free electrons the ion doppler frequency shift due to spheromak particle motion should be four fold larger and comparison of ion doppler and Thomson Scattering data should give clear results.
Note that in principle this technique can also be used to determine the dominant ion mass in a compressed spheromak. The spheromak ion velocity Vi measured via the Ion Doppler technique is inversely proportional to the ion mass.
In executing the above experimental measurements it is essential to remember that the Thomson scattering and ion doppler measurement techniques were originally developed for random plasmas, but the particle motion in a spheromak is not random.
The motion path of the electrons and ions is a spiral within the plasma sheet and has a substantial axial component within the spheromak core. These issues will seriously affect the calibration of some instruments. Also both Thomson Scattering and Ion Doppler instrumentation sample the plasma density near the core of the spheromak where the spacial density of free electrons and ions is much higher than the average spacial density in the spheromak. Failure to properly take these issues into account can lead to a computed average free electron/ion density that is much higher than the actual average free electron/ion density. Moreover, this erroneous calculation may appear to be confirmed by data from an ion probe, which will also indicate too high due to sampling the outer surface of the spheromak.
This web page last updated April 2, 2015.