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Published by Oak Ridge National Laboratory
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Editor: James A. Rome Issue 154 October 2016
E-Mail: jamesrome@gmail.com Phone (865) 482-5643
On the Web at http://web.ornl.gov/info/stelnews/
Multiregion relaxed MHD —
A new approach to an old
quandary
Three-dimensional (3D) MHD equilibria are still an outstanding
theoretical and numerical challenge. The main
reason is that current sheets are predicted to form at rational
surfaces in general 3D equilibria with continuously
nested flux surfaces [1]. Of course, the small but finite
plasma resistivity allows these currents to diffuse and tear
the magnetic field lines, thus forming magnetic islands
around resonant rational surfaces. When islands are large
enough to overlap, magnetic chaos emerges. Also,
depending on the physical mechanisms at play, island selfhealing
can occur, in which case localized plasma currents
can be sustained.
Independent of what mechanisms are at play in determining
the saturated size of magnetic islands, the question of
how to compute the equilibrium magnetic field that is consistent
with the established equilibrium pressure profile is
still under debate [2–5]. In fact, it is an outstanding challenge
to compute 3D MHD equilibria —which generally
consist of an intricate combination of flux surfaces,
islands, and chaos—in a fast, robust, and verifiable fashion.
A recently developed theory based on a generalized
energy principle, referred to as Multiregion, Relaxed
MHD (MRxMHD), was developed [6] and elegantly
bridges the gap between Taylor’s relaxation theory and
ideal MHD. MRxMHD allows for partial relaxation
(which may lead to development of islands) and incorporates
the possibility of non-smooth solutions (that describe
current sheets).
The Stepped-Pressure Equilibrium Code (SPEC), a nonlinear
implementation of MRxMHD, was developed in the
last few years at PPPL [5]. SPEC has been benchmarked
against VMEC in the axisymmetric case [7]; it is the first
code ever tp compute 3D MHD equilibria with nested surfaces
and the predicted singular current densities [8]; it
was used to reproduce self-organized helical states in
reversed field pinches [9]; and it has been used to study
the response to resonant magnetic perturbations, with and
without islands [5,10,11]. SPEC is clearly a candidate to
compute 3D MHD stellarator equilibria with current
sheets and magnetic islands, and has recently been brought
from PPPL to IPP Greifswald in order to explore stellarator
equilibria.
The SPEC code calculates MHD equilibria as extrema of
the MRxMHD energy functional. In MRxMHD, the
plasma is partitioned into a finite number, N, of nested volumes,
Vv , v = 1, …, N, that undergo Taylor relaxation.
These volumes are separated by N - 1 interfaces, Iv , that
are constrained to remain magnetic surfaces during the
energy minimization process. The location and shape of
these surfaces are a priori unknown and determined selfconsistently
by a force-balance condition. The MRxMHD
equilibrium states satisfy:
in
in
where [[ ]] is the jump across the vth interface and p is
the plasma pressure, which is constant in each relaxed volume.
While for N = 1 the theory trivially reduces to Taylor’s
theory, it has been shown that in the formal limit N 
 ideal MHD is exactly retrieved [7].
  B = B V
p B2
2
+ ----- = 0 I 
In this issue . . .
Three-dimensional equilibria are essential to properly
analyze stellarators. The Stepped-Pressure Equilibrium
Code (SPEC) has been developed at PPPL, and
as a first step towards predictive capability, SPEC has
been verified for stellarator vacuum fields including
islands. The next steps to be undertaken are the calculation
of stellarator equilibria with finite prescribed
current and pressure in free boundary. ................... 1
Stellarator News -2- October 2016
As of now, SPEC is a fixed-boundary code and requires
specification of the boundary in terms of the harmonics of
its geometry. Akin to equilibrium codes, SPEC also needs
specification of two profiles, e.g., the pressure in each
relaxed volume, p(v), and the rotational transform on
either side of each interface, (v), in terms of the toroidal
magnetic flux, v , enclosed in each volume. The SPEC
code provides a solution for the magnetic field and the
shape and location of the ideal interfaces.
As a first step towards predictive capability, SPEC has
been verified for stellarator vacuum fields including
islands. In fact, a vacuum field can be understood as a single
Taylor state with  = 0. Thus SPEC can be used with N
= 1,  = 0 (no current), and p = 0 (no pressure).
As an example, Fig.1 shows the magnitude of the magnetic
field obtained from SPEC on the outer surface of the
Wendelstein 7-X (W7-X) limiter vacuum configuration,
which includes a 5/6 island resonance. The solution has
been exactly verified by showing that the volume-averaged
error converges exponentially towards machine precision
as the Fourier resolution is increased [12]. Also, a
rigorous benchmark with Biot-Savart solutions has been
successfully carried out (see, e.g., Fig. 2).
Finally, multivolume calculations, namely with N > 1,
have also been verified, thus providing confidence that
SPEC is correctly (i.e., with arbitrary accuracy) calculating
MRxMHD equilbrium states.
This verification exercise has also motivated the search for
faster algorithms that can maintain the current computation
speed at high resolution in strongly shaped configurations
such as W7-X.
The next steps to be undertaken are the calculation of stellarator
equilibria with finite prescribed current and pressure
in free boundary, which should provide insights into
(1) the effect of bootstrap current on the formation of
islands and (2) the equilibrium limit, which is related to
the emergence of magnetically stochastic regions.
Joaquim Loizu
Max Planck Institute for Plasma Physics
Greifswald, Germany
References
[1] P. Helander, Rep. Prog. Phys. 77(8), 087001 (2014).
[2] A. H. Reiman and A. H. Greenside, Comput. Phys.
Commun. 43, 157 (1986)
[3] Y. Suzuki, N. Nakajima, K. Watanabe, Y. Nakamura
and T. Hayashi, Nucl. Fusion 46, L19 (2006).
[4] S. P. Hirshman, R. Sanchez and C. R. Cook, Phys. Plasmas
18, 062514 (2011).
[5] S. R. Hudson et al, Phys. Plasmas 19, 112502 (2012).
[6] M. J. Hole, S. R. Hudson and R. L. Dewar, Nucl. Fusion
47(8), 746–753 (2007).
[7] G. R. Dennis, S. R. Hudson, R. L. Dewar and M. J.
Hole, Phys. Plasmas 20(3), 032509 (2013).
[8] J. Loizu, S. Hudson, A. Bhattacharjee and P. Helander,
Phys. Plasmas 22(2), 022501 (2015).
[9] G. R. Dennis et al, Phys. Rev. Lett. 111, 055003 (2013).
[10] J. Loizu, S. Hudson, A. Bhattacharjee, S. Lazerson and
P. Helander, Phys. Plasmas 22(9), 090704 (2015).
[11] J. Loizu, S. Hudson, P. Helander, S. Lazerson, and A.
Bhattacharjee, Phys. Plasmas 23(5), 055703 (2016).
[12] J. Loizu, S. Hudson, and C. Nührenberg, submitted to
Phys. Plasmas (2016).
Fig. 1. Amplitude of the magnetic field on the W7-X boundary
magnetic surface. Results obtained from SPEC.
Fig. 2. Profile of the rotational transform obtained from
field-line tracing on the SPEC and Biot-Savart solutions for
the vacuum field in the W7-X limiter configuration.