A stepped Beltrami field model for internal transport barriers

(a)M. J. Hole*, (b)S. R. Hudson and (a)R. L. Dewar
 (a) Res. School of Phys. Sciences and Eng., Australian National University, ACT 0200, Australia
(b) PPPL, P.O. Box 451, Princeton, New Jersey 08543, U.S.A.
matthew.hole@anu.edu.au

A stepped Beltrami plasma model comprises regions in which the magnetic field structure is Beltrami () and the pressure gradient zero.  Different plasma regions are normally separated by an ideal interface, across which there can be a sheet current and change in pressure and current density. 

In solar plasmas, Beltrami field models have long been used to model coronal fields.  Over the last few years however, Beltrami and stepped Beltrami field models have received more attention in the magnetic confinement fusion community.  Recently, a Beltrami model was used to describe Edge Localised Modes, which are cyclic disturbances in the outer annular plasma region of a tokamak.  A Taylor relaxation model, similar to that discussed here, predicts the characteristic width of plasma affected by ELM’s [1].  A stepped Beltrami plasma model has also been proposed to describe cyclic quasi-helicity states in the reverse field pinch [2]

Recently, Hole et al [4] proposed stepped Beltrami plasmas as an analytically well-founded mathematical framework to describe 3D plasma equilibria.  In fully 3D configurations, the magnetic field does not in general form flux surfaces – rather, the field lines chaotically fill a volume.  Such fields, which are in general Beltrami, offer a sound mathematical basis upon which to describe general 3D plasma equilibria [5].  Hole et al [4] have introduced a variational model, and solved for the equilibria and stability in cylindrically periodic plasmas. 

We build on this description to investigate the stability of internal transport barriers in tokamak plasmas.  A transport barrier is a step change in confinement, and leads to a step change in pressure.  Transport barriers only exist in plasmas in which the safety factor, a measure of the local field pitch in the plasma is either flat, or weakly reverse shear in the core. 

We postulate a two interface model for the transport barrier.  Deploying our variational model [4], we scan stability over the minimum value of the safety factor, and the value of the central safety factor.  We appeal to poloidal and toroidal discretisation of wave modes to remove unstable modes from a periodic cylinder. Finally, we show stable configurations do exist, thereby offering a possible energetic motivation for the formation of internal transport barriers in reverse shear plasmas. 

[2] C. G. Gimblett et al, Phys. Rev. Lett. , 96, 035006, 2006.
[3] E. Tassi et al, Phys. Plas., 14, 092109, 2007
[4] M.J. Hole et al, Nuc. Fus., 47, 746, 2007
[5] S. R. Hudson et al. Phys. Plas. 72, 1167, 2007