Taylor state explained

In plasma physics, a Taylor state is the minimum energy state of a plasma while the plasma is conserving magnetic flux.[1] This was first proposed by John Bryan Taylor in 1974 and he backed up this claim using data from the ZETA machine.[2]

Taylor-States are critical to operating both the Dynomak and the reversed field pinch - both run in a Taylor State.

Examples

In 1974, Dr. John B Taylor proposed that a spheromak could be formed by inducing a magnetic flux into a loop plasma. The plasma would then relax naturally into a spheromak also known as a Taylor State.[3] [4] This process worked if the plasma:

These claims were later checked by Marshall Rosenbluth in 1979.[5] In 1974, Dr. Taylor could only use results from the ZETA pinch device to back up these claims. But, since then, Taylor states have been formed in multiple machines including:

Derivation

Consider a closed, simply-connected, flux-conserving, perfectly conducting surface

S

surrounding a plasma with negligible thermal energy (

\beta0

).

Since

\vec{B}\vec{ds}=0

on

S

. This implies that

\vec{A}||=0

.

As discussed above, the plasma would relax towards a minimum energy state while conserving its magnetic helicity. Since the boundary is perfectly conducting, there cannot be any change in the associated flux. This implies

\delta\vec{B}\vec{ds}=0

and

\delta\vec{A}||=0

on

S

.

We formulate a variational problem of minimizing the plasma energy

W=\intd3rB

2/2\mu
\circ
while conserving magnetic helicity

K=\intd3r\vec{A}\vec{B}

.

The variational problem is

\deltaW\deltaK=0

.

After some algebra this leads to the following constraint for the minimum energy state

\nabla x \vec{B}=λ\vec{B}

.

Notes and References

  1. Book: Paul M. Bellan. 2000. Spheromaks: A Practical Application of Magnetohydrodynamic dynamos and plasma self-organization. 71–79. Imperial College Press. 978-1-86094-141-2.
  2. Taylor, J. Brian. "Relaxation of toroidal plasma and generation of reverse magnetic fields." Physical Review Letters 33.19 (1974): 1139.
  3. Bellan, Paul (2000). Spheromaks. Imperial College Press. ISBN 978-1-86094-141-2.
  4. Taylor, J. Brian. "Relaxation of toroidal plasma and generation of reverse magnetic fields." Physical Review Letters 33.19 (1974): 1139.
  5. Rosenbluth, M. N., and M. N. Bussac. "MHD stability of spheromak." Nuclear Fusion 19.4 (1979): 489
  6. JARBOE, T. R., WYSOCKI, F.J., FERNÁNDEZ, J.C., HENINS, I., MARKLIN, G.J., Phys. Fluids B 2 (1990) 1342-1346
  7. "Physics through the 1990s", National Academies Press, 1986, p. 198.
  8. WYSOCKI, F.J., FERNÁNDEZ, J.C., HENINS, I., JARBOE, T.R., MARKLIN, G.J., Phys. Rev. Letters 21 (1988) 2457-2460
  9. Wood, R. D., et al. "Particle control in the sustained spheromak physics experiment." Journal of nuclear materials 290 (2001): 513-517.
  10. Sieck, P. E., et al. "First Plasma Results from the HIT-SI Spheromak." APS Division of Plasma Physics Meeting Abstracts. Vol. 45. 2003.
  11. Sutherland, D. A., et al. "The dynomak: An advanced fusion reactor concept with imposed-dynamo current drive and next-generation nuclear power technologies."